# Thread:

1. ## x(u,v),y(u,v) to u(x,y),v(x,y)

Can anybody help with this:
I have $x=u+sin v, y=v+cos u$. How to extract u,v as functions from x,y, i.e. to get u(x,y) and v(x,y)?

2. Essentially, you want to solve the two non-linear equations x= u+ sin v, y= v+ cos u for u and v in terms of x and y. I doubt that there will be any elementary solution for that.

3. yes, i believe in that, so i am looking for some particular step which could make this task easier.

4. Does a problem specifically ask for u(x,y) and v(x,y)?

Because if it is only asking for derivatives at certain points, you don't actually need the explicit formulas.

5. No, the task actually is:
$x=u+sin(v)$
$y=v+cos(u)$
$z(u,v)=uv^2$
Find $\frac{\partial z}{\partial x} + \frac {\partial z}{\partial y}$ at point $(\frac {\pi +3}{3} , \frac {\pi +1}{3}).$
So I must find $\frac{\partial z}{\partial u} \frac{\partial u}{\partial x}$, $\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}$, $\frac{\partial z}{\partial v} \frac{\partial v}{\partial x}$, $\frac{\partial z}{\partial v} \frac{\partial v}{\partial y}$.

6. Right, so you don't actually need the explicit formula.

For example, to find $\frac{\partial u}{\partial x}$ and $\frac{\partial v}{\partial x}$.

Take the partials of the equations directly (independent variables are $x,y$):
$<br /> \frac{\partial x}{\partial x} = \frac{\partial u}{\partial x} + \cos(v)\frac{\partial v}{\partial x}<br />$

$<br /> \frac{\partial y}{\partial x} = \frac{\partial v}{\partial x} - \sin(u)\frac{\partial u}{\partial x}<br />$

Note that $\frac{\partial x}{\partial x} = 1$ and $\frac{\partial y}{\partial x}=0$.

Solve this system of linear equations at the given point for $\frac{\partial v}{\partial x}, \frac{\partial u}{\partial x}$.

Similar process for other derivatives.

Note you do need to solve for u and v at the point $(\frac {\pi +3}{3} , \frac {\pi +1}{3})$, and you use these values when evaluating things like sin(u).

7. Originally Posted by waytogo
No, the task actually is:
$x=u+sin(v)$
$y=v+cos(u)$
$z(u,v)=uv^2$
Find $\frac{\partial z}{\partial x} + \frac {\partial z}{\partial y}$ at point $(\frac {\pi +3}{3} , \frac {\pi +1}{3}).$
So I must find $\frac{\partial z}{\partial u} \frac{\partial u}{\partial x}$, $\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}$, $\frac{\partial z}{\partial v} \frac{\partial v}{\partial x}$, $\frac{\partial z}{\partial v} \frac{\partial v}{\partial y}$.
Do you know about Jacobians? I would suggest you check the point. It looks a little dubious.

8. Is $\left(\frac{\pi+ 3}{3}, \frac{\pi+ 1}{3}\right)$ (x, y) or (u, v)?

9. the point was incorrect, the actual point is $(\frac{\pi+3}{3},\frac{\pi+1}{2})$.
If I have caught your idea, I should use theorem about the inverse Jacobian?

10. The point now makes sense. Yes, inverse Jacobians is the way I would do this.
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