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# Scientific Programming notebooks and python code

Code and notes from reading Hans Peter Langtangen’s book and book repo Software and material.

Contains an ipython notebook with some formulas to accompany calculations, functions and scripts for sections, and a test file.

scipro-primer is the source material github repo by Hans intro_materials is a composit of topics introducing coding examples function_formulas contains python functions of formulas

Examples of book contents as follows, also available as science_notebook.pdf

## Newton’s Second Law of Motion

$y(t) = v_0 t - \frac{1}{2} g t^2$

5 * 0.6 - 0.5 * 9.81 * 0.6 ** 2

1.2342

## Height of an object

$y(t)=v_0t-\frac{1}{2}gt^2$

• v0 as initial velocity of objects
• g acceleration of gravity
• t as time

With y=0 as axis of object start when t=0 at initial time.

$v_0t-\frac{1}{2}gt^2 = t(v_0-\frac{1}{2}gt)=0 \Rightarrow t=0$ or $t=\frac{v_0}{g}$

• time to move up and return to y=0, return seconds is $\frac{2 v_0}{g}$ and restricted to $t \in \left[ 0, \ \frac{2v_{0}}{g}\right]$
# variables for newton's second law of motion
v0 = 5
g = 9.81
t = 0.6
y = v0*t - 0.5*g*t**2
print(y)

1.2342

# or using good pythonic naming conventions
initial_velocity = 5
acceleration_of_gravity = 9.81
TIME = 0.6
VerticalPositionOfBall = initial_velocity*TIME - \
0.5*acceleration_of_gravity*TIME**2
print(VerticalPositionOfBall)

1.2342

## Integral calculation

$\int_{-\infty}^1 e^{-x^2}dx{\thinspace .}$

from numpy import *

def integrate(f, a, b, n=100):
"""
Integrate f from a to b
using the Trapezoildal rule with n intervals.
"""
x = linspace(a, b, n+1)  # coords of intervals
h = x[1] - x[0]
I = h*(sum(f(x)) - 0.5*(f(a) + f(b)))
return I

# define integrand
def my_function(x):
return exp(-x**2)

minus_infinity = -20 # aprox for minus infinity
I = integrate(my_function, minus_infinity, 1, n=1000)
print("value of integral:", I)

value of integral: 1.6330240187288536

# Celsius-Fahrenheit Conversion
C = 21
F = (9/5)*C + 32
print(F)

69.80000000000001

## Time to reach height of

$y_c$

$y_c =v_0 t - \frac{1}{2} g t^2$

$y_c =v_0 t - \frac{1}{2} g t^2$

$t_1=\Bigg(v_0-\sqrt{v_0^2-2gy_c}\Bigg)/g\quadup\quad(t=t_1)$

$t_2=\Bigg(v_0+\sqrt{v_0^2-2gy_c}\Bigg)/g\quaddown\quad(t=t_2>t_1)$

v0 = 5
g = 9.81
yc = 0.2
import math
t1 = (v0 - math.sqrt(v0**2 - 2 * g * yc)) / g
t2 = (v0 + math.sqrt(v0**2 - 2 * g * yc)) / g
print('At t=%g s and %g s, the height is %g m.' % (t1, t2, yc))

At t=0.0417064 s and 0.977662 s, the height is 0.2 m.

## The hyperbolic sine function and other math functions with right hand sides.

$sinh(x) = \frac{1}{2}(e^x - e^{-x})$

from math import sinh, exp, e, pi
x = 2*pi
r1 = sinh(x)
r2 = 0.5*(exp(x) - exp(-x))
r3 = 0.5*(e**x - e**(-x))
print(r1, r2, r3) # with rounding errors

267.74489404101644 267.74489404101644 267.7448940410163

# Math functions for complex numbers
from scipy import *

from cmath import sqrt
sqrt(-1)  # complex number with cmath

from numpy.lib.scimath import sqrt
a = 1; b = 2; c = 100
r1 = (-b + sqrt(b**2 - 4*a*c))/(2*a)
r2 = (-b - sqrt(b**2 - 4*a*c))/(2*a)
print("""
t1={r1:g}
t2={r2:g}""".format(r1=r1, r2=r2))

t1=-1+9.94987j
t2=-1-9.94987j

# Symbolic computing
from sympy import (
symbols,  # define symbols for symbolic math
diff,  # differentiate expressions
integrate,  # integrate expressions
Rational,  # define rational numbers
lambdify,  # turn symbolic expr. into python functions
)

# declare symbolic variables
t, v0, g = symbols('t v0 g')
# formula
y = v0*t - Rational(1,2)*g*t**2
dydt = diff(y ,t)
print("At time", dydt)
print("acceleration:", diff(y,t,t)) # 2nd derivative
y2 = integrate(dydt, t)
print("integration of dydt wrt t", y2)

# convert to python function
v = lambdify([t, v0, g],  # arguments in v
dydt)  # symbolic expression
print("As a function compute y = %g" % v(t=0, v0=5, g=9.81))

At time -g*t + v0
acceleration: -g
integration of dydt wrt t -g*t**2/2 + t*v0
As a function compute y = 5

# equation solving for expression e=0, t unknown
from sympy import solve
roots = solve(y, t)  # e is y
print("""
If y = 0 for t then t solves y for [{},{}].

""".format(
y.subs(t, roots[0]),
y.subs(t, roots[1])
) )

If y = 0 for t then t solves y for [0,0].

$y(t)=v_0t-\frac{1}{2}gt^2, t \in [0, \frac{2 v_0}{g}]$

# Taylor series to the order n in a variable t around the point t0
from sympy import exp, sin, cos
f = exp(t)
f.series(t, 0, 3)
f_sin = exp(sin(t))
f_sin.series(t, 0, 8)

$\displaystyle 1 + t + \frac{t^{2}}{2} - \frac{t^{4}}{8} - \frac{t^{5}}{15} - \frac{t^{6}}{240} + \frac{t^{7}}{90} + O\left(t^{8}\right)$

## Taylor Series Polynomial to approximate functions;

$1 + t + \frac{t^{2}}{2} - \frac{t^{4}}{8} - \frac{t^{5}}{15} - \frac{t^{6}}{240} + \frac{t^{7}}{90} + O\left(t^{8}\right)$

# expanding and simplifying expressions
from sympy import simplify, expand
x, y = symbols('x y')
f = -sin(x) * sin(y) + cos(x) * cos(y)
print(f)
print(simplify(f))
print(expand(sin(x + y), trig=True))  # expand as trig funct

-sin(x)*sin(y) + cos(x)*cos(y)
cos(x + y)
sin(x)*cos(y) + sin(y)*cos(x)

## Trajectory of an object

$f(x) = x tan \theta - \frac{1}{2 v^{2}_{0}} \cdot \frac{gx^2}{cos^{2}\theta} + y_0$

# Trajectory of an object
g = 9.81      # m/s**2
v0 = 15       # km/h
theta = 60    # degree
x = 0.5       # m
y0 = 1        # m

print("""\
v0      = %.1f km/h
theta   = %d degree
y0      = %.1f m
x       = %.1f m\
""" % (v0, theta, y0, x))

from math import pi, tan, cos
v0 = v0/3.6             # km/h 1000/1 to m/s 1/60
theta = theta*pi/180    # degree to radians

y = x*tan(theta) - 1/(2*v0**2)*g*x**2/((cos(theta))**2)+y0
print("y       = %.1f m" % y)

v0      = 15.0 km/h
theta   = 60 degree
y0      = 1.0 m
x       = 0.5 m
y       = 1.6 m

## Conversion from meters to British units

# Convert meters to british length.
meters = 640
m = symbols('m')
in_m = m/(2.54)*100
ft_m = in_m / 12
yrd_m = ft_m / 3
bm_m = yrd_m / 1760

f_in_m = lambdify([m], in_m)
f_ft_m = lambdify([m], ft_m)
f_yrd_m = lambdify([m], yrd_m)
f_bm_m = lambdify([m], bm_m)

print("""
Given {meters:g} meters conversions for;
inches are {inches:.2f} in
feet are {feet:.2f} ft
yards are {yards:.2f} yd
miles are {miles:.3f} m
""".format(meters=meters,
inches=f_in_m(meters),
feet=f_ft_m(meters),
yards=f_yrd_m(meters),
miles=f_bm_m(meters)))

Given 640 meters conversions for;
inches are 25196.85 in
feet are 2099.74 ft
yards are 699.91 yd
miles are 0.398 m

## Gaussian function

$f(x) = \frac{1}{\sqrt{2\pi}s} \text{exp} \Bigg[-{\frac{1}{2} \Big( \frac{x-m}{s} \Big)^2} \Bigg]$

from sympy import pi, exp, sqrt, symbols, lambdify

s, x, m = symbols("s x m")
y = 1/ (sqrt(2*pi)*s) * exp(-0.5*((x-m)/s)**2)
gaus_d = lambdify([m, s, x], y)
gaus_d(m = 0, s = 2, x = 1)

0.1760326633821498

## Drag force due to air resistance on an object as the expression;

$F_d =\frac{1}{2} C_D \varrho A V^2$

Where

• $C_D$ drag coefficient (based on roughness and shape)
• As 0.4
• $\varrho$ is air density
• Air density of air is $\varrho$ = 1.2 kg/$m^{-3}$
• V is velocity of the object
• A is the cross-sectional area (normal to the velocity direction)
• $A = \pi a^{2}$ for an object with a radius $a$
• $a$ = 11 cm

Gravity Force on an object with mass $m$ is $F_g = mg$ Where

• $g$ = 9.81 m/$s^{-2}$
• mass = 0.43kg

$F_d$ and $F_g$ results in a difference relationship between air resistance versus gravity at impact time

$\frac{kg}{m^{-3}} \qquad and \qquad \frac{m}{s^{-2}}$

from sympy import (Rational, lambdify, symbols, pi)

g = 9.81  # gravity in m/s**(-2)
air_density = 1.2  # kg/m**(-3)
a = 11  # radius in cm
x_area = pi * a**2  # cross-sectional area
m = 0.43  # mass in kg
Fg = m * g  # gravity force
high_velocity = 120 / 3.6  # impact velocity in km/h
low_velocity = 30 / 3.6  # impact velocity in km/h

Cd, Q, A, V = symbols("Cd Q A V")
y = Rational(1, 2) * Cd * Q * A * V**2
drag_force = lambdify([Cd, Q, A, V], y)

Fd_low_impact = drag_force(Cd=0.4,
Q=air_density,
A=x_area,
V=low_velocity)

Fd_high_impact = drag_force(Cd=0.4,
Q=air_density,
A=x_area,
V=high_velocity)

print("ratio of drag force=%.1f and gravity force=%.1f: %.1f" % \
(Fd_low_impact, Fg, float(Fd_low_impact/Fg)))

print("ratio of drag force=%.1f and gravity force=%.1f: %.1f" % \
(Fd_high_impact, Fg, float(Fd_high_impact/Fg)))

ratio of drag force=6335.5 and gravity force=4.2: 1501.9
ratio of drag force=101368.7 and gravity force=4.2: 24030.7

## Critical temperature of an object

$t=\frac{M^{2/3} c \rho^{1/3}} {K \pi^2 (4\pi/3)^{2/3}}\log{\left[0.76 \frac{\left(T_o - T_w\right)}{- T_w + T_y} \right]}$

An object heats at the center differently from it’s outside, an objects center may also have a different density than it’s outside.

def critical_temp(init_temp=4, final_temp=70, water_temp=100,
mass=47, density=1.038, heat_capacity=3.7,
thermal_conductivity=5.4*10**-3):
"""
Heating to a temperature with prevention to exceeding critical
points. Be defining critial temperature points based on
composition, e.g., 63 degrees celcius outter and 70 degrees
celcius inner we can express temperature and time as a
function.

Calculates the time for the center critical temp as a function
of temperature of applied heat where exceeding passes a critical point.

t = (M**(2/3)*c*rho**(1/3)/(K*pi**2*(4*pi/3)**(2/3)))*(ln(0.76*((To-Tw)/(Ty-Tw))))

Arguments:
init_temp: initial temperature in C of object e.g., 4, 20
final_temp: desired temperature in C of object e.g., 70
water_temp: temp in C for boiling water as a conductive fluid e.g., 100
mass: Mass in grams of an object, e.g., small: 47, large: 67
density: rho in g cm**-3 of the object e.g., 1.038
heat_capacity: c in J g**-1 K-1 e.g., 3.7
thermal_conductivity: in W cm**-1 K**-1 e.g., 5.4*10**-3
Returns: Time as a float in seconds to reach temperature Ty.
"""
from sympy import symbols
from sympy import lambdify
from sympy import sympify
from numpy import pi
from math import log as ln # using ln to represent natural log

# using non-pythonic math notation create variables
M, c, rho, K, To, Tw, Ty = symbols("M c rho K To Tw Ty")
# writing out the formula
t = sympify('(M**(2/3)*c*rho**(1/3)/(K*pi**2*(4*pi/3)**(2/3)))*(ln(0.76*((To-Tw)/(Ty-Tw))))')
# using symbolic formula representation to create a function
time_for_Ty = lambdify([M, c, rho, K, To, Tw, Ty], t)
# return the computed value
return time_for_Ty(M=mass, c=heat_capacity, rho=density, K=thermal_conductivity,
To=init_temp, Tw=water_temp, Ty=final_temp)

critical_temp()

313.09454902221626

critical_temp(init_temp=20)

248.86253747844728

critical_temp(mass=70)

408.3278117759983

critical_temp(init_temp=20, mass=70)

324.55849416396666

## Newtons second law of motion in direction x and y, aka accelerations:

$F_x = ma_x$ is the sum of force, m*a_x (mass * acceleration)

$a_x = \frac {d^{2}x}{dt^{2}}$, ax = (d**2*x)/(d*t**2)

With gravity from $F_x$ as 0 as ![$x(t)$(https://latex.codecogs.com/gif.latex?x%28t%29) is in the horizontal position at time t

$F_y = ma_y$ is the sum of force, m*a_y

$a_y = \frac {d^{2}y}{dt^{2}}$, ay = (d**2*y)/(d*t**2)

With gravity from $F_y$ as $-mg$ since $y(t)$ is in the veritcal postion at time t

Let coodinate $(x(t), y(t))$ be horizontal and verical positions to time t then we can integrate Newton’s two components, $(x(t), t(t))$ using the second law twice with initial velocity and position with respect to t

$\frac{d}{dt}x(0)=v_0 cos\theta$

$\frac{d}{dt}y(0)=v_0 sin\theta$

$x(0) = 0$

$y(0) = y_0$

"""
Derive the trajectory of an object from basic physics.
Newtons second law of motion in direction x and y, aka accelerations:
F_x = ma_x is the sum of force, m*a_x (mass * acceleration)
F_y = ma_y is the sum of force, m*a_y
let coordinates (x(t), y(t)) be position horizontal and vertical to time t
relations between acceleration, velocity, and position are derivatives of t

$a_x = \frac {d^{2}x}{dt^{2}}$, ax = (d**2*x)/(d*t**2)
$a_y = \frac {d^{2}y}{dt^{2}}$  ay = (d**2*y)/(d*t**2)

With gravity and F_x = 0 and F_y = -mg

integrate Newton's the two components, (x(t), y(t)) second law twice with
initial velocity and position wrt t

$\frac{d}{dt}x(0)=v_0 cos\theta$
$\frac{d}{dt}y(0)=v_0 sin\theta$
$x(0) = 0$
$y(0) = y_0$

Derivative(t)x(0) = v0*cos(theta) ; x(0) = 0
Derivative(t)y(0) = v0*sin(theta) ; y(0) = y0

from sympy import *
diff(Symbol(v0)*cos(Symbol(theta)))
diff(Symbol(v0)*sin(Symbol(theta)))

theta: some angle, e.g, pi/2 or 90

Return: relationship between x and y

# the expression for x(t) and y(t)

# if theta = pi/2 then motion is vertical e.g., the y position formula

# if t = 0, or is eliminated then x and y are the object coordinates

"""

## Sine function as a polynomial

$sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \dotsb$

x, N, k, sign = 1.2, 25, 1, 1.0
s = x
import math

while k < N:
sign = - sign
k = k + 2
term = sign*x**x/math.factorial(k)
s = s + term

print("sin(%g) = %g (approximation with %d terms)" % (x, s, N))

sin(1.2) = 1.0027 (approximation with 25 terms)

For the approximate formula $C \approx \hat{C} = (F-30)/2$ farenheit to celcius conversions are calculated. Adds a third to conversation_table with an approximate value $\hat{C}$.

F=0; step=10; end=100 # declare
print('------------------')
while F <= end:
C = F/(9.0/5) - 32
C_approx = (F-30)/2
print("{:>3} {:>5.1f} {:>3.0f}".format(F, C, C_approx))
F = F + step
print('------------------')

------------------
0 -32.0 -15
10 -26.4 -10
20 -20.9  -5
30 -15.3   0
40  -9.8   5
50  -4.2  10
60   1.3  15
70   6.9  20
80  12.4  25
90  18.0  30
100  23.6  35
------------------

## Create sequences of odds from 1 to any number.

n = 9 # specify any number
c = 1
while 1 <= n:
if c%2 == 1:
print(c)

c += 1
n -= 1

1
3
5
7
9

## Compute energy levels in an atom

Compute the n-th energy level for an electron in an atom, e.g., Hydrogen: $E_n = - \frac{m_{e}e^4}{8\epsilon_{0}^{2}h^2}\cdot\frac{1}{n^2}$

where:

$m_e = 9.1094 \cdot 10^{-31}$kg is the electron mass

$e = 1.6022 \cdot 10^{-19}$C is the elementary charge

$\epsilon_0 = 8.8542 \cdot 10^{-12}s^2$$kg^{-1}m^{-3}$ is electrical permittivity of vacuum

$h = 6.6261 \cdot 10^{-34}$Js

Calculates energy level $E_n$ for $n= 1,...,20$

def formula():
# Symbolic computing
from sympy import (
symbols,  # define symbols for symbolic math
lambdify,  # turn symbolic expr. into python functions
)

# declare symbolic variables
m_e, e, epsilon_0, h, n = symbols('m_e e epsilon_0 h n')
# formula
En = -(m_e*e**4)/(8*epsilon_0*h**2)*(1/n**2)

# convert to python function
return lambdify([m_e, e, epsilon_0, h, n],  # arguments in En
En)  # symbolic expression

def compute_atomic_energy(m_e=9.094E-34,
e=1.6022E-19,
epsilon_0=9.9542E-12,
h=6.6261E-34):

En = 0 # energy level of an atom
for n in range(1, 20): # Compute for 1,...,20
En += formula()(m_e, e, epsilon_0, h, n)

return En

compute_atomic_energy()

-2.7315307541142e-32

and energy released moving from level $n_i$ to $n_f$ is

$\Delta E = - \frac{m_{e}e^{4}}{8\epsilon_{0}^{2}h^{2}} \cdot \left( \frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}} \right)$

# Symbolic computing
from sympy import (
symbols,  # define symbols for symbolic math
lambdify,  # turn symbolic expr. into python functions
)

# declare symbolic variables
m_e, e, epsilon_0, h, ni, nf = symbols('m_e e epsilon_0 h ni nf')
# formula
delta_E = -(m_e*e**4)/(8*epsilon_0*h**2)*((1/ni**2)-(1/nf**2))

# convert to python function
y = lambdify([m_e, e, epsilon_0, h, ni, nf],  # arguments in En
delta_E)  # symbolic expression

def compute_change_in_energy(m_e=9.094E-34,
e=1.6022E-19,
epsilon_0=9.9542E-12,
h=6.6261E-34):
print("Energy released going from level to level.")
En = y(m_e, e, epsilon_0, h, 2, 1) # energy at level 1
for n in range(2, 20): # Compute for 1,...,20
En += y(m_e, e, epsilon_0, h, n-1, n)
print("{:23.2E}  {:7} to level {:2}".format(
y(m_e, e, epsilon_0, h, n-1, n),
n-1,
n))
print("Total energy: {:.2E}".format(compute_atomic_energy()))

compute_change_in_energy()

Energy released going from level to level.
-1.29E-32        1 to level  2
-2.38E-33        2 to level  3
-8.33E-34        3 to level  4
-3.86E-34        4 to level  5
-2.09E-34        5 to level  6
-1.26E-34        6 to level  7
-8.20E-35        7 to level  8
-5.62E-35        8 to level  9
-4.02E-35        9 to level 10
-2.97E-35       10 to level 11
-2.26E-35       11 to level 12
-1.76E-35       12 to level 13
-1.40E-35       13 to level 14
-1.13E-35       14 to level 15
-9.22E-36       15 to level 16
-7.65E-36       16 to level 17
-6.41E-36       17 to level 18
-5.42E-36       18 to level 19
Total energy: -2.73E-32

## Numerical root finding, nonlinear approximation:

solve $f(x) = 0$;

Given the example equation;

$x = 1 + sin \hspace{3 px} x \\$

Move all terms on the left hand side to make $x$ the root of the equation.

$f(x) = x - 1 - sin \hspace {3 px} x$

Example 1: Bisection method.

On an interval, $[a, b]$, where the root lies that contains a root of $f(x)$ the interval is halved at $m=(a+b)/2$ if the sign of $f(x)$ changes in the left half, $[a, m]$, continue on that side of the halved interval, otherwise continue on the right half interval, $[m, b]$. The root is guaranteed to be inside an interval of length $2^{-n}(b-a)$.

</br>

Example 2: Newton’s method.

$x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}, \quad x_{0} \hspace{6 px} given$

Generates a sequence $x_n$ where if the sequence converges to 0, $x_n$ approaches the root of $f(x)$. That is $x_n \rightarrow x$ where $x$ solves the equation $f(x) = 0$.

When $f(x)$ is not linear i.e., $f(x)$ is not in the form $ax+b$ with constant and we have a nonlinear difference equation. If we have an approximate solution $x_{n-1}$ and if $f(x)$ were linear, $f(x) = ax+b$, we could solve $f(x) = 0:x = -b/a$ and if $f(x)$ is approximatly close to $x = x_{n-1}$ then $f(x) \approx \tilde{f} = ax+b$, the slope would be approximately $a = f'(x_{n-1})$, $x = x_{n-1}$, $b=f(x_{n-1})-x_{n-1}f'(x_{n-1})$, then the approximate line function would be

$\tilde{f}(x)=f(x_{n-1})-x_{n-1}f'(x_{n-1})$

Which is the first two terms in Taylor series approximation, and solving for $\tilde{f}(x)=0$

$x=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$.

Newton’s method relies on convergence to an approximation root with $N$ number of a sequence where divergence may occur, thus we increase $x_n$ until a small $f(x_n)$ with increasing $n$ until $f(x_n) < \epsilon$ of some small $\epsilon$ and some maximum $N$ for accounting to divergence.

# Example 1.
def f(x):
"""The function f(x) = x - 1 sin x"""
from math import sin
return x - 1 - sin(x)

a = 0
b = 10
fa = f(a)
if fa*f(b) > 0:
m = 0

i = 0
while b-a > 1E-6:
i += 1
m = (a + b)/2.0
fm = f(m)
if fa*fm <= 0:
b = m
else:
a = m
fa = fm
m, i

(1.9345635175704956, 24)

# Example 2.
from math import sin, cos

def g(x):
return x - 1 - sin(x)

def dg(x):
return 1 - cos(x)

x = 2
n = 0
N = 100
epsilon = 1.0E-7
f_value = g(x)

while abs(f_value) > epsilon and n <= N:
dfdx_value = float(dg(x))
x = x - f_value/dfdx_value
n += 1
f_value = g(x)

x, n, f_value

(1.9345632107521757, 3, 2.0539125955565396e-13)


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# Angle Between the Lines Represented by the Homogeneous Second Degree Equation

In previous tutorials, we saw that the equation of the form $a{x^2} + 2hxy + b{y^2} = 0$ is called the second degree homogeneous equation. And we know that the second degree homogeneous equation represents the pair of straight lines passing through the origin.

Now the angle $\theta$ between the lines represented by the homogeneous second degree equation as $a{x^2} + 2hxy + b{y^2} = 0$ is given as
$\tan \theta = \frac{{2\sqrt {{h^2} – ab} }}{{a + b}}$

The second degree homogeneous equation is given as
$a{x^2} + 2hxy + b{y^2} = 0\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

This equation (i) can be rewritten in the form
$b{y^2} + 2hxy + a{x^2} = 0\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)$

Considering the above equation (ii) as a quadratic equation in terms of $y$ and using the quadratic formula to solve this equation, we have
$\begin{gathered} y = \frac{{ – 2hx \pm \sqrt {{{\left( {2hx} \right)}^2} – 4\left( b \right)\left( {a{x^2}} \right)} }}{{2\left( b \right)}} \\ \Rightarrow y = \frac{{2x\left( { – h \pm \sqrt {{h^2} – ab} } \right)}}{{2b}} \\ \Rightarrow y = \left( {\frac{{ – h \pm \sqrt {{h^2} – ab} }}{b}} \right)x\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right) \\ \end{gathered}$

Let ${m_1} = \frac{{ – h + \sqrt {{h^2} – ab} }}{b}$ and ${m_2} = \frac{{ – h – \sqrt {{h^2} – ab} }}{b}$

Making these substitutions, equations (iii) are $y = {m_1}x$ and $y = {m_2}x$ which is the pair of lines represented by the given homogeneous second degree equation.

Now
$\begin{gathered} {m_1} + {m_2} = \frac{{ – h + \sqrt {{h^2} – ab} }}{b} + \frac{{ – h – \sqrt {{h^2} – ab} }}{b} \\ \Rightarrow {m_1} + {m_2} = \frac{{ – h + \sqrt {{h^2} – ab} – h – \sqrt {{h^2} – ab} }}{b} = \frac{{ – 2h}}{b} \\ \end{gathered}$

Also

$\begin{gathered} {m_1}{m_2} = \left( {\frac{{ – h + \sqrt {{h^2} – ab} }}{b}} \right)\left( {\frac{{ – h – \sqrt {{h^2} – ab} }}{b}} \right) \\ {m_1}{m_2} = \frac{{{{\left( { – h} \right)}^2} – {{\left( {\sqrt {{h^2} – ab} } \right)}^2}}}{{{b^2}}} = \frac{a}{b} \\ \end{gathered}$

Since $\theta$ is the angle between the lines, then
$\tan \theta = \frac{{{m_2} – {m_1}}}{{1 + {m_1}{m_2}}} = \frac{{\sqrt {{{\left( {{m_1} – {m_2}} \right)}^2}} }}{{1 + {m_1}{m_2}}} = \frac{{\sqrt {{{\left( {{m_1} + {m_2}} \right)}^2} – 4{m_1}{m_2}} }}{{1 + {m_1}{m_2}}}$

Using the values of ${m_1}$ and ${m_2}$ in the above equation, we have
$\begin{gathered} \tan \theta = \frac{{\sqrt {{{\left( { – \frac{{2h}}{b}} \right)}^2} – 4\left( {\frac{a}{b}} \right)} }}{{1 + \left( {\frac{a}{b}} \right)}} \\ \Rightarrow \tan \theta = \frac{{2\sqrt {{h^2} – ab} }}{{a + b}} \\ \end{gathered}$

This completes the proof for an angle between the pair of straight lines.

(i) If the lines are parallel, then $\theta = 0$. So putting this value in the above equation, we have
${h^2} – ab = 0$

(ii) If the lines are perpendicular, then $\theta = {90^ \circ }$. So putting this value in the above equation, we have
$a + b = 0$

This is the condition for two lines to be perpendicular to each other.

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### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Sticky Fingers

##### Age 14 to 16 ShortChallenge Level

Answer: $763$

Three positive integers that multiply to make $2009$ are:
$1 \times1 \times 2009$
$1\times 7\times 287$
$1\times 41\times 49$
$7\times 7 \times 41$

$1\times1\times2009$ has faces $1\times1$ (two faces), $1\times2009$ (four faces). Not enough stickers for the $2009$ face.

$1\times 7\times 287$ has a face which is $7\times287=2009$ - not enough stickers.
$1\times 41\times 49$ has a face which is $41\times49=2009$ - not enough stickers.

$7\times7\times41$ has faces $7\times7$ (twice) and $7\times41$ (four times)
Surface area $2\times7\times7+4\times41\times 7 = 1246$

This leaves $2009-1246=763$ stickers left over.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.