text:
# Developments in Geometric Algebra

This page is a sub-page of our page on New Foundations for Classical Mechanics.

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Related KMR-pages:

• Chapter 1: Origins of Geometric Algebra

• Chapter 3: Mechanics of a Single Particle
• Chapter 4: Central Forces and Two-Particle Systems

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Books:

René Descartes (1637), La Géometrie
Hermann Günther Graßmann (1844), Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik
A New Branch Of Mathematics – The Ausdehnungslehre of 1844 and Other Works, translated by Lloyd C. Kannenberg (1995)
William Kingdon Clifford (1878), Applications of Grassmann’s Extensive Algebra, American Journal of Mathematics Vol. 1, No. 4 (1878), pp. 350-358 (9 pages), Published by: The Johns Hopkins University Press
David Hestenes (1993, (1986)), New Foundations for Classical Mechanics

• Zwikker (1963)

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Other relevant sources of information:

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/////// Quoting Hestenes: New Foundations for Classical Mechanics (Chapter 2):

Chapter 2: Developments in Geometric Algebra

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Table Of Content

2-5.
2-6. Analytic Geometry

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In Chapter 1 we developed geometric algebra as a symbolic system for representing the basic geometrical concepts of direction, magnitude, orientation and dimension. In this chapter we continue the development of geometric algebra into a full-blown mathematical language. The basic grammar of this language is completely specified by the axioms set down at the end of Chapter 1. But there is more to a language than its grammar!

To develop geometric algebra to the point where we can express and explore the ideas of mechanics with fluency, in this chapter we introduce auxiliary concepts and definitions, derive useful algebraic relations, describe simple curves and surfaces with algebraic equations, and formulate the fundamentals of differentiation and integration with respect to scalar variables. Further mathematical developments are given in Chapter 5.

2-1. Basic Identities and Definitions

In Chapter 1 we were led to the geometric product for vectors by combining inner and outer products according to the equation

$\, \bold a \, \bold b = \bold a \cdot \bold b + \bold a \wedge \bold b. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.1)

Then we reversed the procedure, defining the inner and outer products in terms of the geometric product by the equations

$\, \bold a \cdot \bold b = \frac{1}{2} \, (\bold a \, \bold b + \bold b \, \bold a) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.2)

$\, \bold a \wedge \bold b = \frac{1}{2} \, (\bold a \, \bold b - \bold b \, \bold a) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.3)

This did more than reduce two different kinds of multiplication to one. It made possible the formulation of a simple axiom system from which an unlimited number of geometrical relations can be deduced by algebraic manipulation. In this section we aim to improve our skills at carrying out such deductions and establish some widely useful results.

The inner and outer product appear frequently in applications, because they have straightforward geometrical interpretations, as we saw in Chapter 1. For this reason, it is often desirable to operate directly with inner and outer products, even though we regard the geometric product as more fundamental. To make this possible, we need a system of algebraic identities relating inner and outer products. We derive these identities, of course, by using the geometric product and the axioms of geometric algebra set down in Section 1-7.

The different products are most easily related by the equation

$\, \bold a \, {\bold A}_r = \bold a \cdot {\bold A}_r + \bold a \wedge {\bold A}_r, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.4)

which generalizes (1.1) to apply to any $\, r$-blade $\, {\bold A}_r$, and therefor to any $\, r$-vector (= sum of $\, r$-blades) with grade $\, r > 0$. Recall that the corresponding definition of inner and outer products are given by

$\, \bold a \cdot {\bold A}_r = \frac{1}{2} ( \bold a \, {\bold A}_r - (-1)^r \, {\bold A}_r \, \bold a ) = (-1)^{r+1} \, {\bold A}_r \cdot \bold a \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.5)

$\, \bold a \wedge {\bold A}_r = \frac{1}{2} ( \bold a \, {\bold A}_r + (-1)^r \, {\bold A}_r \, \bold a ) = (-1)^r \, {\bold A}_r \wedge \bold a \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.6)

We will make frequent use of the fact that $\, \bold a \cdot {\bold A}_r \,$ is an $\, (r -1)$-vector (which is a scalar if $r = 1$) while $\, \bold a \wedge {\bold A}_r \,$ is an $\, (r + 1)$-vector.

To illustrate the use of (1.4) and its special case (1.1), let us derive the associative rule for the outer product. Beginning with the associative rule for the geometric product,

$\, \bold a \, ( \bold b \, \bold c ) = (\bold a \, \bold b ) \, \bold c )$,

we use (1.1) to get

$\, \bold a \, ( \bold b \cdot \bold c + \bold b \wedge \bold c ) = (\bold a \cdot \bold b + \bold a \wedge \bold b ) \, \bold c )$.

Applying the distributive rule and (1.4), we get

$\, \bold a ( \bold b \cdot \bold c ) + \bold a \cdot ( \bold b \wedge \bold c ) + \bold a \wedge ( \bold b \wedge \bold c ) = ( \bold a \cdot \bold b ) \bold c + ( \bold a \wedge \bold b ) \cdot \bold c + ( \bold a \wedge \bold b ) \wedge \bold c$.

Now we identify the terms $\, \bold a \, ( \bold b \cdot \bold c ) \,$ and $\, \bold a \cdot ( \bold b \wedge \bold c ) \,$ as vectors, and the term
$\, \bold a \wedge ( \bold b \wedge \bold c ) \,$ as a trivector. Since vectors are distinct from trivectors, we can separately equate vector and trivector parts on each side of the equation. By equating trivector parts, we get the associative rule

$\, \bold a \wedge ( \bold b \wedge \bold c ) = ( \bold a \wedge \bold b ) \wedge \bold c. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.7)

And by equating the vector parts we find an algebraic identity which we have not seen before.

$\, \bold a \, ( \bold b \cdot \bold c ) + \bold a \cdot ( \bold b \wedge \bold c ) = ( \bold a \cdot \bold b ) \, \bold c + ( \bold a \wedge \bold b ) \cdot \bold c.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.8)

This derivation of the associative rule (1.7) should be compared with our previous derivation of the same rule in Section 1-6. That derivation was considerably more complicated, because it employed a direct reduction of the outer product to the geometric product. Moreover, the indirect method employed here gives us the additional “vector identity” (1.8) at no extra cost.

Note the general structure of the method: an identity involving geometric products alone is expanded into inner and outer products by using (1.4) and then parts of the same grade are separately equated. This will be our principal method for establishing identities involving inner and outer products. As another example, note that the method immediately gives the distributive rules for inner and outer products. Thus, if $\, \bold a \,$ is a vector and $\, {\bold B}_r \,$ and $\, {\bold C}_r \,$ are $\, r$-blades, then by applying (1.4) to

$\, \bold a \, ( {\bold B}_r + {\bold C}_r) = \bold a \, {\bold B}_r + \bold a \, {\bold C}_r$,

and separating parts of different grade, we get

$\, \bold a \cdot ( {\bold B}_r + {\bold C}_r ) = \bold a \cdot {\bold B}_r + \bold a \cdot {\bold C}_r, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.9a)

and

$\, \bold a \wedge ( {\bold B}_r + {\bold C}_r) = \bold a \wedge {\bold B}_r + \bold a \wedge {\bold C}_r. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.9b)

Here we have the distributive rules in a somewhat more general (hence more useful) form than they were presented in Chapter 1.

These examples show the importance of separating a multivector or a multivector equation into parts of different grades. So it will be useful to introduce a special notation to express such a separation. Accordingly, we write $\, {\langle \, A \, \rangle}_r \,$ to denote the $\, r$-vector part of a multivector $\, A$. For example, if $\, A = \bold a \bold b \bold c$, this notation enables us to write

$\, {\langle \, \bold a \bold b \bold c \, \rangle}_3 \,$

for the trivector part,

$\, {\langle \, \bold a \bold b \bold c \, \rangle}_1 = \bold a \, ( \bold b \cdot \bold c ) + \bold a \cdot ( \bold b \wedge \bold c ) \,$

for the vector part, while the vansishing of scalar and bivector parts is described by the equation

$\, {\langle \, \bold a \bold b \bold c \, \rangle}_0 = 0 = {\langle \, \bold a \bold b \bold c \, \rangle}_2$.

According to axiom (7.1) of Chapter 1, every multivector can be decomposed into a sum of its $\, r$-vector parts, as expressed by writing

$\, A = \sum\limits_{r} {\langle \, A \, \rangle}_r = {\langle \, A \, \rangle}_0 + {\langle \, A \, \rangle}_1 + {\langle \, A \, \rangle}_2 + {\langle \, A \, \rangle}_3 \, . \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.10)

If $\, A = {\langle \, A \, \rangle}_r \,$, then $\, A \,$ is said to be homogeneous of grade $\, r$, that is, $\, A \,$ is an $\, r$-vector. A multivector $\, A \,$ is said to be even (odd) if $\, A = {\langle \, A \, \rangle}_r = 0 \,$ when $\,r \,$ is an odd (even) integer. Obviously every multivector $\, A \,$ can be expressed as a sum of an even part $\, {\langle \, A \, \rangle}_+ \,$ and an odd part $\, {\langle \, A \, \rangle}_- \,$. Thus

$\, A = {\langle \, A \, \rangle}_+ + {\langle \, A \, \rangle}_- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.11a)

where

$\, {\langle \, A \, \rangle}_+ = {\langle \, A \, \rangle}_0 + {\langle \, A \, \rangle}_2 \, , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.11b)

$\, {\langle \, A \, \rangle}_- = {\langle \, A \, \rangle}_1 + {\langle \, A \, \rangle}_3 \, . \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (1.11c)

We shall see later that the distinction between even and odd multivectors is important, because the even multivectors form an algebra by themselves but the odd multivectors do not.

According to (1.10), we have $\, {\langle \, A \, \rangle}_k = 0 \,$ for all $\, k > 3$, that is, every blade with grade $\, k > 3$ must vanish. We adopted this condition in Chapter 1 to express the fact that physical space is three dimensional, so we will be assuming it in our treatment of mechanics throughout this book. However, such a condition is not essential for mathematical reasons, and there are other applications of geometric algebra to physics where it is not appropriate. For the sake of mathematical generality, therefore, all results and definitions in this section are formulated without limitations of grade, with the exception, of course, of (1.10) and (1.11b, c). This generality is achieved at very little extra cost, and it has the advantage of revealing precisely what features of geometric algebra are peculiar to three dimensions.

Before continuing, it will be worthwhile to discuss the use of parentheses in algebraic expressions. Note that the expression $\, \bold a \cdot \bold b \, \bold c \, \,$ is ambiguous. It could mean $\, (\bold a \cdot \bold b) \, \bold c \,$, which is to say that the inner product is $\, \bold a \cdot \bold b \,$ is carried out first and the resulting scalar is multiplied by the vector $\, \bold c$. On the other hand, it could mean $\, \bold a \cdot ( \bold b \, \bold c )$, which is to say that the geometric product $\, \bold b \,\bold c \,$ is performed before the inner product. The two interpretations give completely different algebraic results. to remove such ambiguities without introducing parentheses, we introduce the following precedence convention: If there is ambiguity, indicated inner and outer products should be performed before any adjacent geometric product.

Thus

$\, ( \bold A \wedge \bold B ) \bold C = \, \bold A \wedge \bold B \bold C ≠ \bold A \wedge ( \bold B \bold C ), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.12a)

$\, ( \bold A \cdot \bold B ) \bold C = \, \bold A \cdot \bold B \bold C ≠ \bold A \cdot ( \bold B \bold C ). \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.12b)

This convention eliminates an appreciable number of parentheses, especially in complicated expressions. Other parentheses can be eliminated by the convention that outer products have “preference” over inner products, so

$\, \bold A \cdot ( \bold B \wedge \bold C ) = \, \bold A \cdot \bold B \wedge \bold C ≠ ( \bold A \cdot \bold B ) \bold C. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.13)

The most useful identity relating inner and outer products is, of course, its simplest one:

$\, \bold a \cdot ( \bold b \wedge \bold c ) = \bold a \cdot \bold b \, \bold c - \bold a \cdot \bold c \, \bold b. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.14)

We derived this in Section 1-6 before we had established our axiom system for geometric algebra. Now we can derive it by a simpler method. First we use (1.2) in the form

$\, \bold a \, \bold b = - \bold b \, \bold a + 2 \, \bold a \cdot \bold b \,$ to reorder multiplicative factors as follows:

$\, \bold a \, \bold b \, \bold c = - \bold b \, \bold a \, \bold c + 2 \, \bold a \cdot \bold b \, \bold c = - \bold b \, ( - \bold c \, \bold a + 2 \, \bold a \cdot \bold c ) + 2 \, \bold a \cdot \bold b \, \bold c \,$.

Rearranging terms and using (1.1), we obtain

$\, \bold a \cdot \bold b \, \bold c - \bold a \cdot \bold c \, \bold b = \frac{1}{2} (\bold a \, \bold b \, \bold c) - \bold b \, \bold c \, \bold a) = \frac{1}{2} ( \bold a \, \bold b \wedge \bold c - \bold b \wedge \bold c \, \bold a )$,

which, by (1.5), gives us (1.14) as desired.

By the same method we can derive the more general reduction formula

$\, \bold a \cdot ( \bold b \wedge {\bold C}_r ) = \bold a \cdot \bold b \, {\bold C}_r - \bold b \wedge ( \bold a \cdot {\bold C}_r ) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1.15)

where $\, \bold a \,$ and $\, \bold b \,$ are vectors, while $\, {\bold C}_r\,$ is an $\, r$-blade.

[…]

It would be quite appropriate to refer to (1.15) as the Laplace expansion of the inner product, because of its relation to the expansion of a determinant (See Chapter 5).

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2-2. The Algebra of a Euclidean Plane

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2-3. The Algebra of Euclidean 3-Space

The concept of a 3-dimensional Euclidean Space $\, {\mathscr{E}}_3 \,$ is fundamental to physics, because it provides the mathematical structure for the concept of physical space. Moreover, the properties of physical space are presupposed in every aspect of mechanics, not to mention the rest of physics. For this reason we cultivate the geometric algebra of $\, {\mathscr{E}}_3 \,$ as the basic conceptual tool for representing and analyzing geometrical relations in physics.

We can analyze the algebra of $\, {\mathscr{E}}_3 \,$ in the same way that we analyzed the algebra of $\, {\mathscr{E}}_2$. Let $\, i \,$ be a unit $\, 3$ blade. The set of all vectors $\, \bold x \,$ which satisfy the equation

$\, \bold x \wedge i = 0 \, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (3.1)

is the Euclidean $\, 3$-dimensional vector space $\, {\mathscr{E}}_3$. Scalar multiples of $\, i \,$ are called pseudoscalars of this vector space, and we refer to $\, i \,$ as the unit-pseudoscalar.

Note that the symbol “$i$” is an exception to our convention that $\, k$-blades be represented in boldface type. We make this exception to emphasize the singular importance of $\, i \,$ and to distinguish it from unit $\, 2$-blades which we have represented by $\, \bold i$. The set of all multivectors is the (geometric) algebra of $\, {\mathscr{E}}_3$, and it will be denoted by $\, {\mathscr{G}}_3 \,$ or $\, {\mathscr{G}}_3 (i)$. One way to study the structure of $\, {\mathscr{G}}_3 \,$ is by constructing a basis for the algebra.

[…]

Quaternions are Spinors

We have seen that the algebra of complex numbers appears with a geometric interpretation as the subalgebra $\, {\mathscr{G}}_2^+ \,$ of even multivectors in $\, {\mathscr{G}}_2 \,$. Similarly, we can express $\, {\mathscr{G}}_3 \,$ as the sum of an odd part $\, {\mathscr{G}}_3^- \,$ and an even part $\, {\mathscr{G}}_3^+ \,$, that is

$\, {\mathscr{G}}_3 = {\mathscr{G}}_3^- + \, {\mathscr{G}}_3^+ \,$.

According to (3.10) then, we can write a multivector $\, A \,$ in the form

$\, A = {\langle \, A \, \rangle}_- + \, {\langle \, A \, \rangle}_+ \, , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (3.12a)

\;\;\;\;\;\;\;\;\;

where

$\, {\langle \, A \, \rangle}_- = \bold a + i \, \beta, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (3.12b)

$\, {\langle \, A \, \rangle}_+ = \alpha + i \, \bold b. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (3.12c)

As is easily verified, $\, {\mathscr{G}}_3^+ \,$ is closed under multiplication, so it is a subalgebra of $\, {\mathscr{G}}_3 \, ,$ though $\, {\mathscr{G}}_3^- \,$ is not. For this reason, $\, {\mathscr{G}}_3^+ \,$ is sometimes called the even subalgebra of $\, {\mathscr{G}}_3 \,$. But it may be better to refer to $\, {\mathscr{G}}_3^+ \,$ as the spinor algebra or subalgebra to emphasize the geometric significance of its elements.

Just as every spinor in $\, {\mathscr{G}}_2^+ \,$ represents a rotation-dilation in $\, 2$-dimensions, so every spinor of $\, {\mathscr{G}}_3^+ \,$ represents a rotation-dilation in $\, 3$-dimensions. The representation of rotations by spinors is discussed fully in Chapter 5.

Equation (3.12c) shows that each spinor can be expressed as the sum of a scalar and a bivector. In view of (3.7), then, the four quantities $\, 1 \, , \, {\bold i}_1 \, , \, {\bold i}_2 \, , \, {\bold i}_3 \,$ make up a basis for $\, {\mathscr{G}}_3^+ \,$. Thus $\, {\mathscr{G}}_3^+ \,$ is a linear space of $\, 4 \,$ dimensions. For this reason, the elements of $\, {\mathscr{G}}_3^+ \,$ were called quaternions by William Rowan Hamilton who invented them in 1843 independently of the full geometric algebra from which they arise here. Following Hamilton we may also use the name Quaternion Algebra for $\, {\mathscr{G}}_3^+ \,$.

Quaternions are well known to mathematicians today as the largest possible associative division algebra. But few are aware of how quaternions fit in the more general system of geometric algebra. Some might say that the quaternion algebra is actually distinct from the spinor algebra $\, {\mathscr{G}}_3^+ \,$, that these algebras are not identical but only isomorphic. But such a distinction only serves to complicate matters unneccesarily. The identification of quaternions with spinors is fully justified not only because they have equivalent algebraic properties, but more important, because they have the same geometric significance.

Hamilton’s choice of the name quaternion is unfortunate, for the name merely refers to the comparatively insignificant fact that the quaternions compose a linear space of $\, 4 \,$ dimensions. The name quaternion diverts attention from the key fact that Hamilton had invented a geometric algebra. Hamilton’s work itself shows clearly the crucial role of geometry in his invention. Hamilton was consciously looking for a system of numbers to represent rotations in three dimensions. He was looking for a way to describe geometry by algebra, so he found a geometric algebra.

Hamilton developed his quaternion algebra at about the same time that Hermann Grassmann developed his “algebra of extension” based on the inner and outer products. In spite of the fact that both Hamilton and Grassmann eventually came to know and admire one another’s work, for several decades neither of them could see how their respective geometric algebras were related. It was only late in his life that Grassmann realized that Hamilton’s quaternions can be derived simply bu adding the inner and outer products to get the geometric product $\, \bold a \, \bold b = \bold a \cdot \bold b + \bold a \wedge \bold b$, but it was too late for him to pursue the implications of this insight very far.

At about the same time, the English mathematician W. K. Clifford independently realized that Hamilton and Grassmann were approaching one and the same subject from different points of view. By combining their algebraic ideas, he was led, in 1876, to the geometric product. Unfortunately, death claimed him before he was able to fully delineate the rich mixture of geometric and algebraic ideas he discovered, and no successor appeared to continue his work with the same depth of geometric insight. Consequently, the mathematical world continued to regard Grassmann’s and Hamilton’s algebras as independent systems. Divided, they fell into relative disuse.

Quaternions today reside in a kind of mathematical limbo, because their place in a more general geometric algebra is not recognized. The prevailing attitude toward quaternions is exhibited in a biographical sketch of Hamilton by the late mathematician E. T. Bell. The sketch is titled “An Irish Tragedy”, because for the last twenty years of his life, Hamilton concentrated all his enormous mathematical powers on the study of quaternions in, as Bell would have it, the quixotic belief that quaternions would play a central role in the mathematics of the future. Hamilton’s judgement was based on a new and profound insight into the relation between algebra and geometry. Bell’s evaluation was made by surveying the mathematical literature nearly a century later. But union with Grassmann’s algebra puts quaternions in a different perspective. It may yet prove true that Hamilton looking ahead saw further than Bell looking back.

Clifford may have been the first person to find significance in the fact that two different interpretations of number can be distinguished, the quantitative and the operational. On the first interpretation, number is a measure of “how much” of “how many” of something. On the second, number describes a relation between different quantities. The distinction is nicely illustrated by recalling the interpretation already given to a unit bivector $\, \bold i$. Interpreted quantitatively, $\, \bold i \,$ is a measure of directed area. Operationally interpreted, $\, \bold i \,$ specifies a rotation in the $\, \bold i$-plane.

Clifford observed that Grassmann developed the idea of directed number from the quantitative point of view, while Hamilton emphasized the operational interpretation. The two approaches are brought together by the geometric product. Either a quantitative or an operational interpretation can be given to any number, yet one or the other may be more important in most applications. Thus, vectors are usually interpreted quantitatively, while spinors are usually interpreted operationally. Of course the algebraic properties of vectors and spinors can be studied abstractly with no reference whatsoever to interpretation. But interpretation is crucial when algebra functions as a language.

The Vector Cross Product

Vector algebra, as conceived by J. Willard Gibbs in 1884, is widely used as the basic mathematical language in physics textbooks today, so it is important to show that this system fits naturally into $\, {\mathscr{G}}_3 \,$. The demonstration is easy. We need only introduce the vector cross product $\bold a \times \bold b \,$ defined by the equation

$\, \bold a \times \bold b = -i \, \bold a \wedge \bold b. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (3.13)

Thus, $\, \bold a \times \bold b \,$ is the vector dual to the bivector $\, \bold a \wedge \bold b$. As shown in Figure 3.3, the sign of the duality is chosen so that the vectors $\, \bold a, \bold b, \bold a \times \bold b$, in that order, form a righthanded set. This agrees with our convention for the handedness of the pseudoscalar $\, i$, for comparing (3.13) with (3.6), we see that

$\, {\sigma}_1 \times {\sigma}_2 = {\sigma}_3. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (3.14)

To remember the correct sign in the duality relation (3.13), it is helpful to note that the geometric product of vectors in $\, {\mathscr{G}}_3 \,$ can be written

$\, \bold a \, \bold b = \bold a \cdot \bold b + \bold a \wedge \bold b = \bold a \cdot \bold b + i \, \bold a \times \bold b, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (3.15)

and (3.13) can be obtained from this by separately equating bivector parts. Finally, note that by squaring (3.13) we deduce

$\, ( \bold a \times \bold b )^2 = -(\bold a \wedge \bold b)^2 = | \, \bold a \wedge \bold b \, |^2. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (3.16)

Hence, the magnitude $\, (\bold a \times \bold b) \,$ is equal to the area of the parallelogram in Figure 3.3, which was identified as $\, | \, \bold a \times \bold b \, | \,$ in Section 1-5.

At this point, a caveat is in order. Books on vector algebra commonly make a distinction between polar vectors and axial vectors, with $\, \bold a \times \bold b \,$ identified as an axial vector, if $\, \bold a \,$ and $\, \bold b \,$ are polar vectors. This confusing practice of admitting two kinds of vectors is wholly unnecessary. An “axial vector” is nothing more than a bivector disguised as a vector. So with bivectors at our disposal, we can do without axial vectors. As we have defined it in (3.13), the quantity $\, \bold a \times \bold b \,$ is a vector in exactly the same sense that $\, \bold a \,$ and $\, \bold b \,$ are vectors.

The ease with which conventional vector algebra fits into $\, {\mathscr{G}}_3 \,$ is no accident. Gibbs constructed his system from the same ideas of Grassmann and Hamilton that have gone into geometric algebra. By the end of the 19th century a lively controversy had developed as to which system was more suitable for the work of theoretical physics, the quaternions or vector algebra. A glance at modern textboks shows that the votaries of vectors were victorious. However, quaternions have reappeared disguised as matrices and proved to be essential in modern quantum mechanics. The ironic thing about the vector-quaternion controversy is that there was nothing substantial to dispute. Far from being in opposition, the two systems complement each other and, as we have seen, are perfectly united in the geometric algebra $\, {\mathscr{G}}_3 \,$. The whole controversy was founded on the failure of everyone involved to appreciate the distinction between vectors and bivectors. Indeed, the word “vector” was originally coined by Hamilton for what we now call a bivector. Gibbs changed the meaning of the word to its present sense, but no one at the time understood the real significance of the change he had made.

2-4. Directions, Projections and Angles

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[…]

2-6. Analytic Geometry

This section can be skipped or lightly perused by readers who are in a hurry to get on with mechanics. It is included here as a reference on elementary concepts and results of Analytic Geometry expressed in terms of geometric algebra.

Analytic Geometry is concerned with the description or, if you will, the representation of geometric curves and surfaces by algebraic equations. The traditional approach to Analytic geometry is accurately called Coordinate Geometry, because it represents each geometrical point by a set of scalars called its coordinates. Curves and surfaces are then represented by algebraic equations for the coordinates of their points.

A major drawback of Coordinate Geometry is the fact that coordinates carry superfluous information which often entails unnecessary complications. Thus, rectangular coordinates $\, (x, y, z) \,$ of a point specify the distance of the point from the three coordinate planes, the coordinate $\, z$, for example, specifying the distance from the $\, (x, y)$-plane. Therefore, equations for a geometric figure in rectangular coordinates describe the relation of that figure to three arbitrarily chosen planes. Obviously, it would be more efficient to describe the figure in terms of its intrinsic properties alone, without introducing extrinsic relations to lines or planes which are frequently of no interest. Geometric algebra makes this possible.

In the language of geometric algebra, each geometrical point is represented or labelled by a vector. Indeed, for mathematical purposes it is often simplest to regard the point and the vector that labels it as one and the same. Of course, we can label a given point by any vector we please, and problems can often be simplified by a judicious selection of the point to be labelled by the zero vector. But, as a rule, once a labelling has been selected, it is unnecessary to change it.

The distinction between a point and its vector label becomes important when geometric algebra is used as a language, for then the point, which is undefined as a mathematical entity, might be identified with a mark on a piece of paper or a “place” among physical objects. However, the vector label retains its status as a purely mathematical entity, and geometric algebra precisely describes its geometric properties. It will be noticed also in the following that, although some vectors designate (or are designated as) points, other vectors describe relations between points or have some other geometrical significance.

The simplest relation between two points $\, \bold a \,$ and $\, \bold b \,$ is the vector $\, \bold a - \bold b$, which, for want of a standard terminology, we propose to call the chord from $\, \bold b \,$ to $\, \bold a$. The magnitude of the chord, $\, | \, \bold a - \bold b \, |$, is called the (Euclidean) distance between $\, \bold b \,$ and $\, \bold a$. The zero vector designates a point called the origin. Since $\, \bold a - \bold 0 = \bold a$, the vector $\, \bold a \,$ specifies both the point $\, \bold a \,$ and the chord from the origin to the point $\, \bold a$, and $\, | \, \bold a \, | = | \, \bold a - \bold 0 \, | \,$ is the distance between the point $\, \bold a \,$ and the origin.

Geometric spaces and figures are sets of points. Euclidean Geometry is concerned with distance relations of the form $\, | \, \bold a - \bold b \, |$ among pairs of points in such spaces and figures. Non-Euclidean geometries are based on alternative definitions of the distance between points, such as $\, \log | \, \bold a - \bold b \, |$.

However interesting it is to explore the implications of alternative definitions of distance, we want our definition to correspond to the relations among physical objects determined by the operational rules for measuring distance, and it is a physical fact that, at least to a high order of approximation, such relations conform to the Euclidean definition of distance. For this reason, we will be concerned with the Euclidean definition of distance only, and the adjective “Euclidean” will be unnecessary.

A set $\, {\mathscr{E}}^n \,$ with elements called points is said to be an $\, n$-dimensional Euclidean Space if it has the following properties:

(1) the points in $\, {\mathscr{E}}^n \,$ can be put in one-to-one correspondence with the vectors in an $\, n$– dimensional vector space. (Each vector then is said to designate or label the corresponding point.
(2) There is a rule for assigning a positive number to every pair of points called the distance between the points, and the points can be labelled in such a way that the distance between any pair of points $\, \bold a \,$ and $\, \bold b \,$ is given by $\, | \, \bold a - \bold b \, | = [ \, (\bold a - \bold b)^2 \, ]^{\frac{1}{2}}$.

Obviously, and vector space can be regarded as a Euclidean space simply by regarding each vector as identical with the point it labels. For most mathematical purposes it is quite sufficient to regard each Euclidean space as a vector space. However, in physical applications it is essential to distinguish between each point and the vector which labels it. This is apparent in the most fundamental application of all, the application of geometry to measurement. .

In Chapter 1 we saw that a complex system of operational rules is needed to determine physical points (i.e. positions of places) and measure distances between them. The set of all physical points determined by these rules is called Physical Space. We saw that Euclidean geometry has certain physical implications when interpreted as a physical theory. We can now completely formulate the physical implications of geometry in the single proposition:

Physical Space is a $\, 3$-dimensional Euclidean Space.

This proposition could be called The Zeroth Law of physics, because it is presumed in the theory of measurement and so in every branch of physics, although the Law must be modified or reinterpreted somewhat to conform to Einstein’s theory of relativity and gravitation. Chapter 9 gives a more complete formulation and discussion of the Zeroth Law in relation to the other laws of mechanics.

We label the points of Physical Space by vectors in the geometric algebra $\, {\mathscr{G}}_3$. These vectors compose a $\, 3$-dimensional Euclidean space $\, {\mathscr{E}}_3 = {\langle {\mathscr{G}}_3 \rangle}_1 \,$ which can be regarded as a mathematical model of Physical Space. The properties of points in $\, {\mathscr{E}}_3$, such as their relations to other points, to lines and to planes, require the complete algebra $\, {\mathscr{G}}_3 \,$ for their description. Since $\, {\mathscr{G}}_3 \,$ thereby provides us with the necessary language to describe relations between points in Physical Space, it is appropriate to call $\, {\mathscr{G}}_3 \,$ the geometric algebra of Physical Space.

The study of curves and surfaces in $\, {\mathscr{E}}_3 \,$ is a purely mathematical enterprise, but its relevance to physics is assured by the correspondence of $\, {\mathscr{E}}_3 \,$ with Physical Space. By appropriate semantic assumptions, a curve in $\, {\mathscr{E}}_3 \,$ can be variously interpreted as the path of a particle, a boundary on a surface or the edge of a solid body.

But in the rest of this section, all such interpretations are deliberately ignored, as we learn to describe the form of curves and surfaces with geometric algebra. Our results can then be used in a variety of physical contexts when we introduce semantic assumptions later on.

Straight Lines

The most basic equations in analytic geometry are those for lines and planes. In Section 2-2 we saw that the equation

$\, \bold x \wedge \bold u = 0 \, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (6.1)

determines a line through the origin when $\, \bold u \,$ is a fixed nonzero vector. The substitution $\, \bold x \rightarrow \bold x - \bold a \,$ in (6.1) has the effect of rigidly displacing each point on the line by the same amount $\, \bold a$. From this we conclude that the line $\, \mathscr{L} = \{ \, \bold x \, \} \,$ with direction $\, \hat{\bold u} \,$ passing through the point $\, \bold a \,$ is determined by the equation

$\, (\bold x - \bold a) \wedge \bold u = 0. \, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.2)

It should be noted that this equation determines the line without reference to any space in which the line might be imbedded, although, of course, we are most interested in lines in $\, {\mathscr{E}}_3$.

Moment and Directance of a Line

Equation (6.2) is a necessary and sufficient condition for a point $\, \bold x \,$ to lie on the line $\, \mathscr{L}$.
All the properties of a line, such as its relations to specified points, lines, and planes can be derived from the defining equation (6.2) by geometric algebra. To see how this can best be done, we derive and study various alternative forms of the defining equation; each reveals a different property of the line. On writing $\, \bold M \,$ for the bivector $\,\bold a \wedge \bold u$, Equation (6.2) takes the form

$\,\bold x \wedge \bold u = \bold M. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (6.3)

Since $\,\bold x \wedge \bold u \wedge \bold u = 0$, multiplication of (6.3) by $\, {\bold u}^{-1} \,$ and use of (1.4) as well as (1.14) yields

$\,(\bold x \wedge \bold u) \cdot {\bold u}^{-1} = \bold x - \bold x \cdot \bold u \, {\bold u}^{-1} ) = \bold M \, {\bold u}^{-1}$.

Hence, for fixed $\, \bold M \,$ and $\, \bold u$,

$\, \bold x = (\bold M + \alpha) \, {\bold u}^{-1} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.4)

is a parametric equation for the line $\, \mathscr{L}$, each point $\, \bold x = \bold x (\alpha) \,$ being determined by a value of the parameter $\, \alpha$.

Introducing the vector

$\, \bold d = \bold M \, {\bold u}^{-1} = \bold M \cdot {\bold u}^{-1}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.5)

Equation (6.4) takes the form

$\, \bold x = \bold d + \alpha \, {\bold u}^{-1}. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.6)

Note that $\, \bold d \,$ is orthogonal to $\, \bold u$, since, by (6.5),

$\, \bold d \cdot \bold u = {\langle \, \bold d \, \bold u \, \rangle}_0 = {\langle \, \bold M \, \rangle}_0 = 0$.

So, by squaring (6.6), one obtains the following expression for the distance
$\, | \, \bold x \, | = | \, \bold x - \bold 0 \, | \,$ between the origin and a point $\, \bold x \,$ on $\, \mathscr{L}$:

$\, {| \, \bold x \, |}^2 = {\bold x}^2 = {\bold d}^2 + {\alpha}^2 \, {\bold u}^{-2}$.

This has its minimum value when $\, \bold x = \bold d$. Thus $\, \bold d \,$ is that point on the line $\, \mathscr{L} \,$ which is “closest” to the origin. We call $\, \bold d \,$ the directance (= directed distance) from the point $\, \bold 0 \,$ to the line $\, \mathscr{L}$. The magnitude $\, | \, \bold d \, | \,$ is the distance from the point $\, \bold 0 \,$ to the line $\, \mathscr{L} \,$ (Figure 6.1).

Fig. 6.1.

By substituting (6.3) into (6.5) we get the useful expression

$\, \bold d = \bold x \wedge \bold u \, {\bold u}^{-1} \equiv P_{\bold u}^{\perp} (\bold x), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,$ (6.7)

where $\, P_{\bold u}^{\perp} \,$ is the rejection operator defined in section 2-4. This tells us how to find the directance of a line from any point of the line.

The bivector $\, \bold M \,$ is called the moment of the line $\, \mathscr{L}$. From (6.5) one finds that $\, | \, \bold d \, | = | \, \bold M \, | \, {| \, {\bold u} \, |}^{-1}$, showing, in particular, that if $\, | \, \bold u \, | = 1$, the magnitude of the moment is equal to the distance from the origin to $\, \mathscr{L}$. From Equation (6.3), it is clear that any oriented line $\, \mathscr{L} \,$ is uniquely determined by specifying its direction $\, \bold u \,$ and its moment $\, \bold M$, or equivalently, the single quantity $\, \bold u + \bold M = (1 + \bold d) \, \bold u$. We shall see that the last way of characterizing a line is useful in rigid body mechanics.

Points on a Line

//// FORTSÄTT HÄR

[…]

Conic Sections

Next to straight lines and circles, the simplest curves are the conic sections, so-called because each can be defined as the intersection of a cone with a plane. We shall prefer the following alternative definition, because it leads directly to a most valuable parametric equation: A conic is the set of all points in the Euclidean plane $\, {\mathscr{E}}_2 \,$ with the property that the distance of each point from a fixed point (the focus) is in fixed ratio (the eccentricity) to the distance of that point from a fixed line (the directrix). To express this as an equation, we will denote the eccentricity by $\, \varepsilon$, the directance from the focus to the directrix by $\, \bold d = d \, \hat{\boldsymbol{\varepsilon}} \,$ with $\, {\hat{\boldsymbol{\varepsilon}}}^2 = 1$, and the directance from the focus to any point on the conic by $\, \bold r \,$ (see Figure 6.10).

Fig. 6.10.

The defining condition for a conic can then be written

$\, \dfrac { | \, \bold r \, | } { d - \bold r \cdot \hat{\boldsymbol{\varepsilon}} } = \varepsilon. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.34)

Solving this for $\, r = | \, \bold r \, | \,$ and introducing the eccentricity vector $\, \boldsymbol{\varepsilon} = \varepsilon \, \hat{\boldsymbol{\varepsilon}} \,$ along with the so-called semi-latus rectum $\, \ell = \varepsilon \, d$, we get the more convenient equation

$\, r = \dfrac { \ell } { 1 + \boldsymbol{\varepsilon} \cdot \hat{\bold r} }.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.35)

This expresses the distance $\, r \,$ from the focus to a point on the conic as a function of the direction $\, \hat{\bold r} \,$ to the point. Alternatively, the same condition can be expressed as a parametric equation for $\, r \,$ as a function of the angle $\, \theta \,$ between $\, \boldsymbol{\varepsilon} \,$ and $\, \hat{\bold r}$. Thus, substituting $\, \boldsymbol{\varepsilon} \cdot \hat{\bold r} = \varepsilon \, \cos \theta \,$ into (6.35), we get

$\, r = \dfrac { \ell } { 1 + \varepsilon \, \cos \theta }.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (6.36)

This is a standard equation for conics, but we usually prefer (6.35), because it shows the dependence of $\, r \,$ on the directions $\, \hat{\boldsymbol{\varepsilon}} \,$ and $\, \hat{\bold r} \,$ explicitly, while this dependence in (6.36) is expressed only indirectly through the definition of $\, \theta$.

Equation (6.35) determines a curve when $\, r \,$ is restricted to directions in a plane, but if $\, r \,$ is allowed to range over all directions of $\, {\mathscr{E}}_3 \,$, then (6.35) describes a 2-dimensional surface called a conicoid. Our definition of a conic can be used for a conicoid simply by interpreting the directrix as a plane instead of a line. Both the conics and the conicoids are classified according to the values of the eccentricity shown in Table 6.1.

Table 6.1.

The 1-parameter family of conics with a common focus and pericenter is illustrated in Figure 6.11. The pericenter is the point on the conic at which $\, r \,$ has a minimum value. In the hyperbolic case there are actually two pericenters, one on each branch of the hyperbola. Only one of these is shown in Figure 6.11. If the conics in Figure 6.11 are rotated about the axis through the focus and the pericenter, they sweep out the corresponding conicoids.

Fig. 6.11 Conics with a common focus and pericenter

The conics and the conicoids have quite a remarkable variety of properties, which is related to the fact that they can be described by many different equations besides (6.35). Rather than undertake a systematic study of those properties, we shall wait for them to arise in the context of physical problems, and we will be better prepared for this when we have the tools of differential calculus at our disposal.

Our study of analytic geometry has just begun. The study of particle trajectories, which we undertake in the next chapter, is largely analytic geometry in $\, {\mathscr{E}}_3$. For those who wish to study the analytic geometry in $\, {\mathscr{E}}_2 \,$ in more detail, the book of Zwikker (1963) is recommended. He formulates analytic geometry in terms of complex numbers and shows how much this improves on the traditional methods of coordinate geometry. Of course, everything he does is easily reexpressed in the language of geometric algebra, which has all the advantages of complex numbers and more.

Indeed, geometric algebra brings further improvements to Zwikker’s treatment by enlarging the algebraic system from $\, {\mathscr{G}}_2^+ \,$ to $\, {\mathscr{G}}_2$, and so introducing the fundamental distinction between vectors and spinors and along with it the concepts of inner and outer products. Most important, geometric algebra provides for the generalization of the geometry in $\, {\mathscr{E}}_2 \,$ to $\, {\mathscr{E}}_3$. The present book develops all the principles and techniques needed for analytic geometry, but Zwikker’s book is a valuable storehouse of particular facts about curves in $\, {\mathscr{E}}_2$. Among other things, it includes the remarkable proof that conic sections as defined by (6.34) really are sections of a cone.

/// FORTSÄTT HÄR

/////// End of quote from Hestenes


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## 3.2 Instantaneous Velocity and Speed

### Learning Objectives

By the end of this section, you will be able to:

• Explain the difference between average velocity and instantaneous velocity.
• Describe the difference between velocity and speed.
• Calculate the instantaneous velocity given the mathematical equation for the velocity.
• Calculate the speed given the instantaneous velocity.

We have now seen how to calculate the average velocity between two positions. However, since objects in the real world move continuously through space and time, we would like to find the velocity of an object at any single point. We can find the velocity of the object anywhere along its path by using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later chapters.

### Instantaneous Velocity

The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply velocity. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is $\overset{\text{–}}{v}=\frac{x({t}_{2})-x({t}_{1})}{{t}_{2}-{t}_{1}}$. To find the instantaneous velocity at any position, we let ${t}_{1}=t$ and ${t}_{2}=t+\text{Δ}t$. After inserting these expressions into the equation for the average velocity and taking the limit as $\text{Δ}t\to 0$, we find the expression for the instantaneous velocity:

$v(t)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{x(t+\text{Δ}t)-x(t)}{\text{Δ}t}=\frac{dx(t)}{dt}.$
Instantaneous Velocity

The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t:

$v(t)=\frac{d}{dt}x(t).$

Like average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point ${t}_{0}$ is the rate of change of the position function, which is the slope of the position function $x(t)$ at ${t}_{0}$. (Figure) shows how the average velocity $\overset{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}$ between two times approaches the instantaneous velocity at ${t}_{0}.$ The instantaneous velocity is shown at time ${t}_{0}$, which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times, ${t}_{1},{t}_{2}$, and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.

Figure 3.6 In a graph of position versus time, the instantaneous velocity is the slope of the tangent line at a given point. The average velocities $\overset{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{{x}_{\text{f}}-{x}_{\text{i}}}{{t}_{\text{f}}-{t}_{\text{i}}}$ between times $\text{Δ}t={t}_{6}-{t}_{1},\text{Δ}t={t}_{5}-{t}_{2},\text{and}\,\text{Δ}t={t}_{4}-{t}_{3}$ are shown. When $\text{Δ}t\to 0$, the average velocity approaches the instantaneous velocity at $t={t}_{0}$.

### Example

#### Finding Velocity from a Position-Versus-Time Graph

Given the position-versus-time graph of (Figure), find the velocity-versus-time graph.

Figure 3.7 The object starts out in the positive direction, stops for a short time, and then reverses direction, heading back toward the origin. Notice that the object comes to rest instantaneously, which would require an infinite force. Thus, the graph is an approximation of motion in the real world. (The concept of force is discussed in Newton’s Laws of Motion.)

#### Strategy

The graph contains three straight lines during three time intervals. We find the velocity during each time interval by taking the slope of the line using the grid.

#### Significance

During the time interval between 0 s and 0.5 s, the object’s position is moving away from the origin and the position-versus-time curve has a positive slope. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is +1 m/s, as shown in (Figure). In the subsequent time interval, between 0.5 s and 1.0 s, the position doesn’t change and we see the slope is zero. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is −0.5 m/s. The object has reversed direction and has a negative velocity.

### Speed

In everyday language, most people use the terms speed and velocity interchangeably. In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar.

We can calculate the average speed by finding the total distance traveled divided by the elapsed time:

$\text{Average speed}=\overset{\text{–}}{s}=\frac{\text{Total distance}}{\text{Elapsed time}}.$

Average speed is not necessarily the same as the magnitude of the average velocity, which is found by dividing the magnitude of the total displacement by the elapsed time. For example, if a trip starts and ends at the same location, the total displacement is zero, and therefore the average velocity is zero. The average speed, however, is not zero, because the total distance traveled is greater than zero. If we take a road trip of 300 km and need to be at our destination at a certain time, then we would be interested in our average speed.

However, we can calculate the instantaneous speed from the magnitude of the instantaneous velocity:

$\text{Instantaneous speed}=|v(t)|.$

If a particle is moving along the x-axis at +7.0 m/s and another particle is moving along the same axis at −7.0 m/s, they have different velocities, but both have the same speed of 7.0 m/s. Some typical speeds are shown in the following table.

Speeds of Various Objects*Escape velocity is the velocity at which an object must be launched so that it overcomes Earth’s gravity and is not pulled back toward Earth.
Speed m/s mi/h
Continental drift ${10}^{-7}$ $2\,×\,{10}^{-7}$
Brisk walk 1.7 3.9
Cyclist 4.4 10
Sprint runner 12.2 27
Rural speed limit 24.6 56
Official land speed record 341.1 763
Speed of sound at sea level 343 768
Space shuttle on reentry 7800 17,500
Escape velocity of Earth* 11,200 25,000
Orbital speed of Earth around the Sun 29,783 66,623
Speed of light in a vacuum 299,792,458 670,616,629

### Calculating Instantaneous Velocity

When calculating instantaneous velocity, we need to specify the explicit form of the position function x(t). For the moment, let’s use polynomials $x(t)=A{t}^{n}$, because they are easily differentiated using the power rule of calculus:

$\frac{dx(t)}{dt}=nA{t}^{n-1}.$

The following example illustrates the use of (Figure).

### Example

#### Instantaneous Velocity Versus Average Velocity

The position of a particle is given by $x(t)=3.0t+0.5{t}^{3}\,\text{m}$.

1. Using (Figure) and (Figure), find the instantaneous velocity at $t=2.0$ s.
2. Calculate the average velocity between 1.0 s and 3.0 s.

Strategy(Figure) gives the instantaneous velocity of the particle as the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we can use (Figure), the power rule from calculus, to find the solution. We use (Figure) to calculate the average velocity of the particle.

#### Solution

1. $v(t)=\frac{dx(t)}{dt}=3.0+1.5{t}^{2}\,\text{m/s}$.Substituting t = 2.0 s into this equation gives $v(2.0\,\text{s})=[3.0+1.5{(2.0)}^{2}]\,\text{m/s}=9.0\,\text{m/s}$.
2. To determine the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of x(1.0 s) and x(3.0 s):
$x(1.0\,\text{s})=[(3.0)(1.0)+0.5{(1.0)}^{3}]\,\text{m}=3.5\,\text{m}$
$x(3.0\,\text{s})=[(3.0)(3.0)+0.5{(3.0)}^{3}]\,\text{m}=22.5\,\text{m.}$

Then the average velocity is

$\overset{\text{–}}{v}=\frac{x(3.0\,\text{s})-x(1.0\,\text{s})}{t(3.0\,\text{s})-t(1.0\,\text{s})}=\frac{22.5-3.5\,\text{m}}{3.0-1.0\,\text{s}}=9.5\,\text{m/s}\text{.}$

#### Significance

In the limit that the time interval used to calculate $\overset{\text{−}}{v}$ goes to zero, the value obtained for $\overset{\text{−}}{v}$ converges to the value of v.

### Example

#### Instantaneous Velocity Versus Speed

Consider the motion of a particle in which the position is $x(t)=3.0t-3{t}^{2}\,\text{m}$.

1. What is the instantaneous velocity at t = 0.25 s, t = 0.50 s, and t = 1.0 s?
2. What is the speed of the particle at these times?

#### Strategy

The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. We use (Figure) and (Figure) to solve for instantaneous velocity.

#### Significance

The velocity of the particle gives us direction information, indicating the particle is moving to the left (west) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can understand these concepts visually (Figure). In (a), the graph shows the particle moving in the positive direction until t = 0.5 s, when it reverses direction. The reversal of direction can also be seen in (b) at 0.5 s where the velocity is zero and then turns negative. At 1.0 s it is back at the origin where it started. The particle’s velocity at 1.0 s in (b) is negative, because it is traveling in the negative direction. But in (c), however, its speed is positive and remains positive throughout the travel time. We can also interpret velocity as the slope of the position-versus-time graph. The slope of x(t) is decreasing toward zero, becoming zero at 0.5 s and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and help interpret the calculations.

Figure 3.9 (a) Position: x(t) versus time. (b) Velocity: v(t) versus time. The slope of the position graph is the velocity. A rough comparison of the slopes of the tangent lines in (a) at 0.25 s, 0.5 s, and 1.0 s with the values for velocity at the corresponding times indicates they are the same values. (c) Speed: $|v(t)|$ versus time. Speed is always a positive number.

The position of an object as a function of time is $x(t)=-3{t}^{2}\,\text{m}$. (a) What is the velocity of the object as a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at t = 1.0 s?

### Summary

• Instantaneous velocity is a continuous function of time and gives the velocity at any point in time during a particle’s motion. We can calculate the instantaneous velocity at a specific time by taking the derivative of the position function, which gives us the functional form of instantaneous velocity v(t).
• Instantaneous velocity is a vector and can be negative.
• Instantaneous speed is found by taking the absolute value of instantaneous velocity, and it is always positive.
• Average speed is total distance traveled divided by elapsed time.
• The slope of a position-versus-time graph at a specific time gives instantaneous velocity at that time.

### Conceptual Questions

There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.

Does the speedometer of a car measure speed or velocity?

If you divide the total distance traveled on a car trip (as determined by the odometer) by the elapsed time of the trip, are you calculating average speed or magnitude of average velocity? Under what circumstances are these two quantities the same?

How are instantaneous velocity and instantaneous speed related to one another? How do they differ?

### Problems

A woodchuck runs 20 m to the right in 5 s, then turns and runs 10 m to the left in 3 s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?

Sketch the velocity-versus-time graph from the following position-versus-time graph.

Sketch the velocity-versus-time graph from the following position-versus-time graph.

Given the following velocity-versus-time graph, sketch the position-versus-time graph.

An object has a position function x(t) = 5t m. (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function.

A particle moves along the x-axis according to $x(t)=10t-2{t}^{2}\,\text{m}$. (a) What is the instantaneous velocity at t = 2 s and t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 s and t = 3 s?

Unreasonable results. A particle moves along the x-axis according to $x(t)=3{t}^{3}+5t\text{}$. At what time is the velocity of the particle equal to zero? Is this reasonable?

### Glossary

instantaneous velocity
the velocity at a specific instant or time point
instantaneous speed
the absolute value of the instantaneous velocity
average speed
the total distance traveled divided by elapsed time

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Ucale

### Course Curriculum

 Kinetic Energy of Rotation Kinetic energy of Rotation 00:14:45 Moment of Inertia Moment of Inertia and its Physical Significance 00:12:35 Radius of Gyration Radius of Gyration 00:13:31 Angular Momentum Angular Momentum and Moment of Inertia 00:13:52 Principle of Conservation of Angular Momentum 00:16:50 Torque Torque 00:17:49 Question:U117PNE6 Unlimited Relation between Torque and Angular Momentum 00:06:01 Theorems on Moment of Inertia Theorem of Parallel Axis 00:19:03 Theorem of Perpendicular Axis 00:18:07 Moment of Inertia of Some Bodies of Regular Shape Moment on Inertia of a Thin Circular Ring 00:02:45 Question:U117PNE12 Unlimited Moment of Inertia of a Uniform Thin Circular Ring abut any Diameter of the Ring 00:02:24 Moment of Inertia of a Uniform Thin Circular Ring about a tangent in the plane of the Ring 00:02:07 Moment of inertia of a Uniform Thin Circular ring about a tangent perpendicular to the plane of the Ring 00:02:09 Moment of inertia of a Uniform Circular Disc 00:01:58 Question:U117PNE10 Unlimited Moment of inertia of a Thin Uniform Rod 00:02:58 Question:U117PNE11 Unlimited Moment of inertia of a Thin Uniform Rod about an axis perpendicular to the length of the rod and passing through one edge 00:02:23 Moment of inertia of a Hollow and Solid Cylinder about its axis 00:03:18 Moment of inertia of a Uniform Hollow Sphere about a Diameter 00:02:16 Moment of inertia of a Uniform Solid Sphere about a diameter 00:02:20 Motion on a Inclined Plane Motion of a cylinder rolling on a inclined plane 00:11:14 Rotational Equilibrium Rotational Equilibrium 00:06:09 Question:U117PNE1 Unlimited Question:U117PNE2 Unlimited Question:U117PNE3 Unlimited Question:U117PNE4 Unlimited Question:U117PNE5 Unlimited Question:U117PNE7 Unlimited Question:U117PNE8 Unlimited Question:U117PNE9 Unlimited Question:U117PNE13 Unlimited Question:U117PNE14 Unlimited Question:U117PNE15 Unlimited Question:U117PNE16 Unlimited NCERT Exercise Solution Question:U117PNS1 Unlimited Question:U117PNS2 Unlimited Question:U117PNS3 Unlimited Question:U117PNS4 Unlimited Question:U117PNS5 Unlimited Question:U117PNS6 Unlimited Question:U117PNS7 Unlimited Question:U117PNS8 Unlimited Question:U117PNS9 Unlimited Question:U117PNS10 Unlimited Question:U117PNS11 Unlimited Question:U117PNS12 Unlimited Question:U117PNS13 Unlimited Question:U117PNS14 Unlimited Question:U117PNS15 Unlimited Question:U117PNS16 Unlimited Question:U117PNS17 Unlimited Question:U117PNS18 Unlimited Question:U117PNS19 Unlimited Question:U117PNS20 Unlimited Question:U117PNS21 Unlimited
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• 研究论文 •

重金属复合污染对眼优角蚱代谢组的影响

1. (1. 河池学院化学与生物工程学院, 广西河池 547000; 2. 广西师范大学生命科学学院, 广西桂林 541000)
• 出版日期:2022-04-20 发布日期:2022-03-24

Effects of combined pollution of heavy metals on the metabolomics of Eucriotettix oculatus (Orthoptera: Tetrigidae)

RONG Wan-Tao1, SONG Zhe1, XIN Lei1, DENG Wei-An2, QIN Yu-Yue1, LI Xiao-Dong1,*

1. (1. College of Chemistry and Bioengineering, Hechi University, Hechi, Guangxi 547000, China; 2. College of Life Sciences, Guangxi Normal University, Guilin, Guangxi 541000, China)
• Online:2022-04-20 Published:2022-03-24

Abstract: 【Aim】 To analyze the contents of heavy metals and metabolomics of Eucriotettix oculatus living in metal and non-metal mining areas, and to explore the effects of combined pollution of heavy metals on the accumulation and metabolomics of heavy metals in E. oculatus. 【Methods】 ICP-MS method was used to determine the content of heavy metals in E. oculatus adults. At the same time, based on UPLC-MS/MS detection platform, selfbuilt database and multivariate statistical analysis, the differences of metabolites in the intestine of E. oculatus adults between the metal mining area and the non-metal mining area were analyzed, and KEGG database was used to conduct annotation and pathway enrichment analysis of the differential metabolites. 【Results】 The contents of nine heavy metals in E. oculatus adults in the metal mining area were 0.4-212.4 times as high as those in the non-metal mining area. The results of multivariate statistical analysis showed that the contents of 112 metabolites changed significantly in the intestine of E. oculatus adults in the metal mining area, mainly including amino acids, fatty acyls, organic acids, nucleotides and benzenes. KEGG annotation and pathway enrichment analysis showed that there were 49 significantly differential metabolites that could be annotated, and 40 significantly differential metabolites were related to metabolic pathways. The most significant enrichment pathways included thyroid hormone synthesis pathway, oxytocin signaling pathway, bile secretion pathway and tyrosine metabolism pathway. 【Conclusion】 There are multiple kinds of heavy metals accumulated in E. oculatus adults living in the environment polluted by multiple heavy metals. Heavy metals can change the composition of metabolites in the intestine of E. oculatus, and the change of some metabolites may be a strategy for E. oculatus to adapt to the habitat polluted by multiple heavy metals.


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### Section 1.3 : Trig Functions

4. Determine the exact value of $$\displaystyle \cos \left( { - \frac{{2\pi }}{3}} \right)$$ without using a calculator.

Show All Steps Hide All Steps

Hint : Sketch a unit circle and relate the angle to one of the standard angles in the first quadrant.
Start Solution

First we can notice that $$- \pi + \frac{\pi }{3} = - \frac{{2\pi }}{3}$$ so (recalling that negative angles rotate clockwise and positive angles rotation counter clockwise) the terminal line for $$- \frac{{2\pi }}{3}$$ will form an angle of $$\frac{\pi }{3}$$ with the negative $$x$$-axis in the third quadrant and we’ll have the following unit circle for this problem.

Hint : Given the obvious symmetry in the unit circle relate the coordinates of the line representing $$- \frac{{2\pi }}{3}$$ to the coordinates of the line representing $$\frac{\pi }{3}$$ and use those to answer the question.
Show Step 2

The line representing $$- \frac{{2\pi }}{3}$$ is a mirror image of the line representing $$\frac{\pi }{3}$$ and so the coordinates for $$- \frac{{2\pi }}{3}$$ will be the same as the coordinates for $$\frac{\pi }{3}$$ except that both coordinates will now be negative. So, our new coordinates will then be $$\left( { - \frac{1}{2}, - \frac{{\sqrt 3 }}{2}} \right)$$ and so the answer is,

$\cos \left( { - \frac{{2\pi }}{3}} \right) = - \frac{1}{2}$


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DAQmx Trigger (Start » Analog Window) (G Dataflow)

Configures the task to start acquiring or generating samples when an analog signal enters or leaves a range you specify.

task in is the name of the task that the operation applies.

source

source is the name of a virtual channel or terminal where there is an analog signal to use as the source of the trigger.

For E Series devices, if you use a channel name, the channel must be the first channel in the task. The only terminal you can use for E Series devices is PFI0.

window top

window top is the upper limit of the window. Specify this value in the units of the measurement or generation.

window bottom

window bottom is the lower limit of the window. Specify this value in the units of the measurement or generation.

error in

error in describes error conditions that occur before this node runs. This input provides standard 'error in' functionality.

when

when specifies whether the task starts measuring or generating samples, when the signal enters the window or when it leaves the window. Use window bottom and window top to specify the limits of the window.

Name Description
Entering Window

Start acquiring or generating samples when the signal enters the window.

Leaving Window

Start acquiring or generating samples when the signal leaves the window.

task out is a reference to the task after this node runs.

error out

error out contains error information. If error in indicates that an error occurred before this node ran, error out contains the same error information. Otherwise, error out describes the error status that this node produces.

Where This Node Can Run:

Desktop OS: Windows

FPGA: LabVIEW NXG does not support FPGA devices

Web Server: Not supported in VIs that run in a web application


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# Why does the error ensemble use the ceiling of the number of classifiers?

What is $$y$$? Why is $$k$$ the ceil of $$n/2$$? What is $$y \geq k$$?

• Where did you get this text/screenshot from? Which book, article, etc.?
– nbro
Dec 4 '20 at 11:19


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Extra Dimensions in String Theory

This is a whole other question that you maybe expect from me, but it is bugging me since I heard of a long time ago. Bosonic string theory (to take the simplest one of string theories) is consistent in only 26 dimensions. This critical dimension is needed to rule out things like negative propabilities, etc.

The way most physisists are calculating the 26 always boggles down to the analytic continuation for the $\zeta(s)$ function that means that $\zeta(-1)=-\tfrac{1}{12}$ and the "definition" $\zeta(-1)=\sum_{k=0}^{\infty} n$. If you don't want to use this, most physisist are relying on a technique called zeta-renormalization, which pretty much looks like the time I set down for my analysis lecture and saw $"\ldots = \tfrac{\infty}{\infty}=1$ on the blackboard (a leftover from a quantum field lecture that wasdn't erased).

Question: Are there mathematically sound proofs for the critical dimensions (for instance, based on some consistency rules of the inproduct on Hilbert spaces) ?

My sincere appologies for bashing theoretical physisists this way, but the leftover was really what I saw (I did even once see $\infty - \infty = 0$ in very much the same context, for that matter).

-
Since there is no axiomatic approach through "constructive quantum field theory" (a la Jaffe and Glimm, etc.) to strings or string field theory, I'd say it's pretty unlikely that you'll find such a mathematically rigorous calculation. –  user02138 Dec 2 '11 at 12:25
I assume you are familiar with Polyakov's "Quantum geometry of bosonic strings" article? The Polyakov formula has been fairly well justified mathematically (and the zeta functional determinant is of great use in studying the spectrum of Riemannian manifolds), so I think your characterisation of zeta-renormalisation is unfair, at least in this particular instance. –  Willie Wong Dec 2 '11 at 14:37
I can't put my hand on the bibliographical details right now, but Floyd Williams of U Mass Amherst has published some rigorous stuff on this question, you might look for that. –  user20911 Dec 9 '11 at 5:15
And as usual, Absence of proof is not a proof of absence. Whatever the context is, it might not be rigorous just yet. As far as I know from my colleagues, most of their time is spent on making sense out of calculations that lead to $\infty$. –  user13838 Dec 9 '11 at 5:39
What I understand back then (1990 or so, when I was studying maths), most physics calculations with infinites in them, which must be right (because they can be shown in labs), are much later rigously proved by mathematicians, not seldom by discovering new areas of maths. –  Willem Noorduin Dec 11 '11 at 20:50


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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Conversion of Customary Units of Measurement

## Convert between customary units for length, mass and volume.

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Conversion of Customary Units of Measurement

Leslie is trying to decide which yogurt to buy at the grocery store. There are two yogurts she is considering. The first kind of yogurt is in a 2 pound container and costs $6. The second kind of yogurt is in a 6 ounce container and is being sold in packs of 5 for$6. She sees that both options cost 6 so she figures she should choose whichever option will give her more yogurt. How can Leslie figure out which option will give her more yogurt? In this concept, you will learn how to convert and compare customary units of measure. ### Converting Customary Units of Measure The customary system of measurement is the system of measurement primarily used in the United States. The tables below show some of the most common U.S. customary units and how they are related.  inch (in) foot (ft) = 12 inches yard (yd) = 3 feet mile (mi) = 5,280 feet  ounce (oz) pound (lb) = 16 ounces  fluid ounce (oz) cup (c) = 8 ounces pint (pt) = 2 cups = 16 ounces quart (qt) = 4 cups = 2 pints = 32 ounces gallon (gal) = 4 quarts  teaspoon (tsp) tablespoon (tbsp) = 3 teaspoons cup (c) = 16 tablespoons To move between different U.S. customary units of length, mass, and capacity, you will multiply or divide by a conversion factor. • Any time you are going from a smaller unit of measure to a larger unit of measure you will need to divide by the conversion factor. • Any time you are going from a larger unit of measure to a smaller unit of measure you will need to multiply by the conversion factor. Here is an example. Convert 374 inches to feet. First, notice that inches are smaller than feet. You are moving from a smaller unit of measure to a larger unit of measure. This means you will need to divide. There are 12 inches in a foot, so your conversion factor is 12. Now, divide. Take the given number of inches and divide by 12. \begin{align*}374 \div 12=\frac{374}{12}=\frac{187}{6}=31 \frac{1}{6}\end{align*} The answer is \begin{align*}374 \text{ inches} = 31 \frac{1}{6} \text{ feet}\end{align*}. Once you know how to convert between different units of measurements, you can compare measurements. The best way to compare two measurements is to first make sure they are in the same unit. Then, you can more easily see which measurement is larger and which is smaller. Let's look at an example. Compare \begin{align*}4 \frac{1}{2}\end{align*} pounds and 74 ounces. First, notice that you have two different units of mass. In order to compare these measurements, convert one of them so that both units are the same. Let's convert \begin{align*}4 \frac{1}{2}\end{align*} pounds to ounces. Now, when converting pounds to ounces you are moving from a larger unit of measure to a smaller unit of measure. This means you will need to multiply. There are 16 ounces in a pound, so your conversion factor is 16. Next, multiply. Take the given number of pounds and multiply by 16. \begin{align*}\begin{array}{rcl} 4 \frac{1}{2} \times 16 = \frac{9}{2} \times \frac{16}{1} = \frac{144}{2} &=& 72 \\ 4 \frac{1}{2} \text{ pounds} &=& 72 \text{ ounces} \end{array}\end{align*} Now you are looking to compare 72 ounces and 74 ounces. 72 ounces is less than 74 ounces, so \begin{align*}4 \frac{1}{2}\end{align*} pounds is less than 74 ounces. The answer is \begin{align*}4 \frac{1}{2} \text{ pounds} < 74 \text{ ounces}\end{align*}. ### Examples #### Example 1 Earlier, you were given a problem about Leslie, who is trying to buy yogurt. She has two options. Her first option is a 2 pound container. Her second option is five 6-ounce containers. Both options will cost her6. She wants to choose the option that will give her more yogurt.

First Leslie should realize that five 6-ounce containers is 30 ounces of yogurt. Leslie needs to compare 2 pounds and 30 ounces. Since Leslie has two different units of mass, in order to compare these measurements she should convert one of them so that both units are the same. She can convert pounds to ounces.

Now, when converting pounds to ounces she is moving from a larger unit of measure to a smaller unit of measure. This means she will need to multiply. There are 16 ounces in a pound, so her conversion factor is 16.

Next, she should multiply. Take the given number of pounds and multiply by 16.

\begin{align*}\begin{array}{rcl} 2 \times 16 &=& 32 \\ 2 \text{ pounds} &=& 32 \text{ ounces} \end{array}\end{align*}

Now Leslie is looking to compare 32 ounces and 30 ounces. 32 ounces is more than 30 ounces, so 2 pounds is greater than 30 ounces.

The answer is Leslie should buy the 2 pound container of yogurt.

#### Example 2

Henrietta is having her 7 best friends over for a luncheon. She wants to prepare salads in which she uses exactly 7 tablespoons of Romano cheese. If she is preparing 8 salads, how many cups of Romano cheese does Henrietta require?

First, figure out how many tablespoons of Romano cheese Henrietta will need. She is making 8 salads and each salad requires 7 tablespoons of cheese.

\begin{align*}8 \times 7 = 56\end{align*}

Henrietta will need 56 tablespoons of Romano cheese.

Now, convert 56 tablespoons to cups. You are moving from a smaller unit of measure to a larger unit of measure. This means you will need to divide. There are 16 tablespoons in a cup, so your conversion factor is 16.

Now, divide.

\begin{align*}56 \div 16 = 3 \frac{1}{2}\end{align*}

The answer is Henrietta will need \begin{align*}3 \frac{1}{2}\end{align*} cups of Romano cheese.

#### Example 3

Convert 82 pints to quarts.

First, notice that pints are smaller than quarts. You are moving from a smaller unit of measure to a larger unit of measure. This means you will need to divide. There are 2 pints in a quart, so your conversion factor is 2.

Now, divide. Take the given number of pints and divide by 2.

\begin{align*}82 \div 2 = 41\end{align*}

The answer is \begin{align*}82 \text{ pints} = 41 \text{ quarts}\end{align*}.

#### Example 4

Compare 80 ounces and 3 quarts.

First, notice that you have two different units of capacity. In order to compare these measurements, convert one of them so that both units are the same. Let's convert 3 quarts to ounces.

Now, when converting quarts to ounces you are moving from a larger unit of measure to a smaller unit of measure. This means you will need to multiply. There are 32 ounces in a quart, so your conversion factor is 32.

Next, multiply. Take the given number of quarts and multiply by 32.

\begin{align*}\begin{array}{rcl} 3 \times 32 &=& 96 \\ 3 \text{ quarts} &=& 96 \text{ ounces} \end{array}\end{align*}

Now you are looking to compare 80 ounces and 96 ounces. 80 ounces is less than 96 ounces, so 80 ounces is less than 3 quarts.

The answer is \begin{align*}80 \text{ ounces} < 3 \text{ quarts}\end{align*}.

#### Example 5

Convert \begin{align*}8 \frac{1}{2}\end{align*} gallons to quarts.

First, notice that gallons are larger than quarts. You are moving from a larger unit of measure to a smaller unit of measure. This means you will need to multiply. There are 4 quarts in a gallon, so your conversion factor is 4.

Now, multiply. Take the given number of gallons and multiply by 4.

\begin{align*}8 \frac{1}{2} \times 4 = \frac{17}{2} \times \frac{4}{1} = \frac{68}{2} = 34\end{align*}

The answer is \begin{align*}8 \frac{1}{2} \text{ gallons} = 34 \text{ quarts}\end{align*}.

### Review

Convert the following measurements into yards. Round to the nearest tenth when necessary.

1. 195 inches
2. 0.2 miles
3. 88 feet
4. 90 feet
5. 900 feet

Convert the following measurements into pounds.

1. 2,104 ounces
2. 96 ounces
3. 3 tons
4. 15 tons

Convert the following measurements into pints.

1. 102 quarts
2. 57 ounces
3. 9.5 gallons
4. 4 quarts
5. 18 quarts
6. 67 gallons

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Customary System

The customary system is the measurement system commonly used in the United States, including: feet, inches, pounds, cups, gallons, etc.

Metric System

The metric system is a system of measurement commonly used outside of the United States. It contains units such as meters, liters, and grams, all in multiples of ten.

Proportion

A proportion is an equation that shows two equivalent ratios.

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.


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Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, March 6, 2014.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2014            March 2014             April 2014
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1                      1          1  2  3  4  5
2  3  4  5  6  7  8    2  3  4  5  6  7  8    6  7  8  9 10 11 12
9 10 11 12 13 14 15    9 10 11 12 13 14 15   13 14 15 16 17 18 19
16 17 18 19 20 21 22   16 17 18 19 20 21 22   20 21 22 23 24 25 26
23 24 25 26 27 28      23 24 25 26 27 28 29   27 28 29 30
30 31

Thursday, March 6, 2014

11:00 am in 241 Altgeld Hall,Thursday, March 6, 2014

#### Partition functions and mock theta functions modulo 2 and 3

###### Scott Ahlgren (UIUC Math)

Abstract: 2 and 3 are often the oddest primes. We will attempt to classify congruences mod 2 and 3 for various partition functions, weakly holomorphic modular forms, and mock theta functions. This generalizes Radu’s recent groundbreaking work on Subbarao’s conjecture for the ordinary partition function. Joint work with B. Kim.

1:00 pm in 347 Altgeld Hall,Thursday, March 6, 2014

#### The Malnormal Special Quotient Theorem

###### Daniel Groves (Illinois-Chicago)

Abstract: Wise's Malnormal Special Quotient Theorem (MSQT) is one of the key technical tools in his work on groups with a quasiconvex hierarchy, and a key ingredient in Agol's proof of the Virtual Haken Conjecture. I will explain the role of the MSQT in this story, and outline a new proof due to Agol, Manning and myself.

2:00 pm in 243 Altgeld Hall,Thursday, March 6, 2014

#### Wolfe's theorem for weakly differentiable cochains

###### Kai Rajala (University of Jyväskylä)

Abstract: The classical Wolfe's theorem isometrically identifies the space of Whitney flat forms with the space of flat cochains (Whitney flat forms are differential forms whose coefficients as well as the coefficients of the exterior derivatives are bounded). We define weakly differentiable cochains using the notion of upper gradient, introduced for functions by Heinonen and Koskela. We then apply these cochains to generalize Wolfe's theorem to the Sobolev space setting. This is joint work with Camille Petit and Stefan Wenger.

2:00 pm in 140 Henry Administration Building,Thursday, March 6, 2014

#### Simple continued fractions with occasional large partial quotients

###### Michael Oyengo (UIUC Math)

Abstract: The continued fraction expansion of certain numbers display surprisingly large partial quotients. We follow the approach by van der Poorten and Shallit of specializing to an integer, the variable in a continued fraction of a Laurent series. It is then easy to tell how and when these large partial quotients occur in the continued fraction expansion of such numbers with a Laurent series representation. We then give well known examples of numbers with this interesting phenomenon.

4:00 pm in 245 Altgeld Hall,Thursday, March 6, 2014

#### String manifolds and loops

###### Richard Melrose (MIT)

Abstract: I will start by discussing the (Whitehead) sequence of successively obstructed structures which may exists on a manifold -- orientation, Spin, String. The analytic difficulties in 'exploiting' the existence of a String structure are substantial and I will discuss an approach with Chris Kottke, which builds on ideas of Atiyah, Witten, Brylinski, Stolz and Teichner and Waldorf, through the loop space of the manifold.


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# c2143 in "yvals.h" ??

This topic is 4024 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

ok so i'm happily coding away...everything is working.. i realise i can make the individual sections of the program (ie: input,graphics etc) seperate. now i have coded this way from the beginning so they are seperate anyway,its more that i realise i can put all the initialisation for them in the respective classes so i do this for the input section (which is using directinput) and bam!..compiler error c2143 in "yvals.h" i have never seen it before and have no idea what is even wrong, let alone how to fix it i looked up yvals.h and as far as i can tell it is a settings file for either VS or the compiler(if they are indeed seperate things) the full error is as follows :
Error	1	error C2143: syntax error : missing ';' before 'namespace'	c:\program files\microsoft visual studio 8\vc\include\yvals.h	480

the code i have is too big to post in its entirety(i think) any hints as to what is wrong or what i've done to break it would be very appreciated also if you need more info just ask and i'll try to post the specific bits of code related Thank you

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Make sure you are not missing a semicolon at the end of any of your class definitions.

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Quote:
 Original post by Driv3MeFarMake sure you are not missing a semicolon at the end of any of your class definitions.

Yep. It's not the problem in yyvals.h, but in one of the files included by that file (written by you), or one that is included somewhere else.

C++ compilers can be really annoying with this particular error message, one took me an hour to find simply due to a stupid syntax error in method definition.

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now i feel a little stupid

usually when i do that(and it happens) i get a different error

thanks for the help, i think that requires a +Rep for you good sir

:)

Edit: *Sirs

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## Electric Field Physics problem

A parallel-plate capacitor is formed from two 1.0  1.0  electrodes spaced2.0  apart. The electric fieldstrength inside the capacitor is .What isthe charge (in ) on positive electrode?Whatis the charge (in ) on negative electrode?


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# logarithms

• January 7th 2007, 08:05 PM
lrn24gve
logarithms
I don't understand this problem:

solve the following system for (x;y):
log x base of 9 + log 8 base of y = 2
log 9 base of x + log y base of 8 = 8/3
• January 7th 2007, 08:41 PM
ThePerfectHacker
Quote:

Originally Posted by lrn24gve
I don't understand this problem:

solve the following system for (x;y):
log x base of 9 + log 8 base of y = 2
log 9 base of x + log y base of 8 = 8/3

$\log_9 x+\log_y 8=2$
$\log_x 9+\log_8 y=8/3$
Multiply the two equations,
$(\log_9 x+\log_y 8)(\log_x 9+\log_8 y)=2(8/3)=16/3$
$\log_9 x\log_x 9+\log_y 8\log_x 9+\log_8 y\log_9 x+\log_y 8\log_8 y=16/3$
$2+\log_y8\log_x 9+\log_8 y\log_9 x=16/3$
$\log_y 8\log_x 9+\log_8 y\log_9 x=10/3$
$z=\log_8 y\log_9 x$
$1/z+z=10/3$
$3+3z^2=10z$
$3z^2-10z+3=0$
$z=\frac{10\pm \sqrt{100-36}}{6}=1/3,3$
The first equation,
$\log_9 x+\frac{1}{\log_8 y}=2$
$A=\log_9 x, B=\log_8 y$
Thus,
$A+\frac{1}{B}=2$
$z=AB=1/3,3$
This can be solved simply.
From the A,B you can find x,y.
• January 7th 2007, 09:23 PM
CaptainBlack
Quote:

Originally Posted by lrn24gve
I don't understand this problem:

solve the following system for (x;y):
log x base of 9 + log 8 base of y = 2
log 9 base of x + log y base of 8 = 8/3

Convert all the logs to a common base ( $e$ - natural logarithms will do) then we have:

$\frac{\ln(x)}{\ln(9)}+\frac{\ln(8)}{\ln(y)}=2$
$\frac{\ln(9)}{\ln(x)}+\frac{\ln(y)}{\ln(8)}=\frac{ 8}{3}$

Which is now a pair of simultaneous equations in $\ln(x)$ and $\ln(y)$ which have to be solved.

RonL
• January 8th 2007, 11:57 AM
Soroban
Hello, lrn24gve!

My solution is not an improvement on the others . . .

Quote:

Solve the following system for $(x,y):$

. . $\begin{array}{cc}\log_9x + \log_y8\:=\:2 \\
\log_x9+ \log_8y \:=\:\frac{8}{3}\end{array}$

Theorem: . $\log_ba \:=\:\frac{1}{\log_ab}$

The equations become:

. . $\begin{array}{cc}\log_9x + \frac{1}{\log_8y} &=\:2 \\ \frac{1}{\log_9x} + \log_8y & =\:\frac{8}{3}\end{array}
\begin{array}{cc}\bf{(1)}\\ \bf{(2)}\end{array}\;\;
\begin{array}{cc}\Rightarrow \\ \Rightarrow\end{array}\;\;
\begin{array}{cc}(\log_9x)(\log_8y) + 1 & =\:2(\log_8y) \\
1 + (\log_9x)(\log_8y) & =\:\frac{8}{3}(\log_8x)\end{array}$

Hence: . $2(\log_8y) \:=\:\frac{8}{3}(\log_9x)\quad\Rightarrow\quad \log_8y \:=\:\frac{4}{3}(\log_9x)\;\;\bf{(3)}$

Subsitute into (1): . $\log_9x + \frac{1}{\frac{4}{3}(\log_9x)} \:=\:2$

. . which simplifies to: . $4(\log_9x)^2 - 8(\log_9x) + 3 \:=\:0$

. . which factors: . $\left[2\log_9x - 1\right]\left[2\log_9x - 3\right] \:=\:0$

And we have two equations to solve:

. . $2\log_9x - 1 \:=\:0\quad\Rightarrow\quad \log_9x = \frac{1}{2}\quad\Rightarrow\quad x \,=\,9^{\frac{1}{2}}\quad\Rightarrow\quad\boxed{x \,=\,3}$
. . . . Substitute into (3): . $\log_8y \,= \,\frac{4}{3}(\log_93) \,=\,\frac{4}{3}\cdot\frac{1}{2}\,=\,\frac{2}{3}$
. . . . . . Then: . $y \,= \,8^{\frac{2}{3}}\quad\Rightarrow\quad\boxed{y\,=\ ,4}$

. . $2\log_9x - 3\:=\:0\quad\Rightarrow\quad \log_9x = \frac{3}{2}\quad\Rightarrow\quad x\,=\,9^{\frac{3}{2}}\quad\Rightarrow\quad\boxed{x \,=\,27}$
. . . . Substitute into (3): . $\log_8y \,=\,\frac{4}{3}(\log_927) \,=\,\frac{4}{3}\cdot\frac{3}{2} \,= \,2$
. . . . . . Then: . $y \,=\,8^2\quad\Rightarrow\quad\boxed{y \,= \,64}$

Solutions: . $(x,y) \:=\:(3,4),\:(27,64)$


text:
# Chapter 2 - Motion Along a Straight Line - Problems - Discussion Questions - Page 58: Q2.22

From a height of $3h$, it would take a time of $\sqrt{3} ~T$ to reach the ground.

#### Work Step by Step

Let $h$ be the height where it was dropped originally. $h = \frac{1}{2}gT^2$ $T = \sqrt{\frac{2h}{g}}$ Now suppose the object is dropped from a height of $3h$. $3h = \frac{1}{2}gt_2^2$ $t_2 = \sqrt{\frac{6h}{g}}$ $t_2 = \sqrt{3}\times \sqrt{\frac{2h}{g}} = \sqrt{3} ~T$ From a height of $3h$, it would take a time of $\sqrt{3} ~T$ to reach the ground.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# $H_0 : \beta_2 \le 0$ or $=0$

This is a part of my lecture note.

To test whether $\beta_2 \le 0$ at the 5% level of significance:

Hypothesis: $H_0 : \beta_2 =0$ and $H_1 : \beta_2 >0$

I don't understand $H_0: \beta_2 =0$ instead of $\beta_2 \le 0$.

My lecturer said it is because the worst case of $\beta_2$ is when it is equal to 0, but I don't really get what it means.

• I don't understand it as well. In a one-sided t-test $H_0$ could be that some parameter ist less or equal $0$. Maybe some context might help. Maybe we are in a situation where we only want to do two.sided testing? Maybe $\beta_2$ is a regression coefficient and our software performs two-sided tests on those and we want to use those? If so, please state in the question. – Bernhard Mar 14 '18 at 10:54
• The idea is that you will reject $H_0$ if the data show enough evidence for a high value of $\beta_2$. In this case you are interested to know if there is enough evidence that $\beta_2$ is larger than 0, which is the upper bound you want to check when testing if $\beta_2 \leq 0$. – Alessandro Mar 14 '18 at 11:45
• Bernhard yes it is a two sided test, and $\beta_2$ is a regression coefficient. Sorry for not mentioning it. – shk910 Mar 14 '18 at 22:41
• Alessandro Thanks, but why we can't say that $H_0: \beta_2 \le 0$ in this case?? My lecturer said $H_0$ must have equality only, but I don't understand. – shk910 Mar 14 '18 at 22:44
• Check this answer: stats.stackexchange.com/questions/7853/… and this: stats.stackexchange.com/questions/8196/… – Alessandro Mar 15 '18 at 11:13


text:
# converting hexadecimal to byte in java

I m new in java.I have a text document with hexadecimal values line by line, i m trying to read it and convert it into byte array. but for the hexadecimal values like 8, d, 11, 0, e4 when parsing i m getting wrong value for e4 as -28 instead of 228. how can i overcome this conversion error....

          FileInputStream fstream = new FileInputStream("C:/Users/data.txt");

DataInputStream in = new DataInputStream(fstream);

byte[] bytes = new byte[1024];
String str;
int i=0;

while ((str = br.readLine()) != null)
{

bytes[i]= (byte) (Integer.parseInt(str,16) & 0xFF);
i++;

}
byte[] destination = new byte[i];
System.arraycopy(bytes, 0, destination, 0, i);

br.close();
return destination;

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byte in Java is signed. It's range is - [-128, 127], the value after 127 goes to the negative side. And hence 228 is -28. –  Rohit Jain Dec 13 '12 at 9:05
Please don't use DataInputStream to read text vanillajava.blogspot.co.uk/2012/08/… –  Peter Lawrey Jan 30 '13 at 20:50

As Alnitak said byte is signed in java. Value of 0xe4 = 228 which is unsigned, and the range of byte is -128 to 127.

My suggestion is to use int instead of byte like

int[] bytes = new int[1024];
bytes[i]= Integer.parseInt(str,16);

You get the same thing which you have require.

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chances are he has a good reason to want an array of bytes (i.e. input to some other API call). The only real reason to convert to an unsigned int is for presentation purposes, which should be separated from the internal representation. –  Alnitak Dec 13 '12 at 9:30
do also note that storing as int will require four times more memory than storing them in a byte array... –  Alnitak Dec 13 '12 at 10:10
@Alnitak Yes you are right ... –  Shantanu Banerjee Dec 13 '12 at 10:18

Bytes (and all other integer types) are signed in Java, not unsigned.

If you're just treating the bytes as a byte array, it doesn't matter that some of the values are negative, their bit representation is still correct.

You can get the "proper" unsigned value by masking the byte value with the int value 0xff, although the resulting value will itselt be an int too:

int n = (myByte & 0xff);

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+1 But it still matters though even if you are treating them as bits and are not just debugging. Why oh why does java have signed... I don't buy the explanation :P –  Esailija Dec 13 '12 at 9:14