Multivariable Calculus Homework

[gunda95]

Homework Statement

See in picture

Homework Equations

What is the final answer?

The Attempt at a Solution

I know dw/du = df/dx * dx/du + df/dy * dy/du

& that dx/du = -8sinu & dy/du = -4sinvsinu

Stumped on how to get df/dx and df/dy

 

[Mark44]

Homework Statement

See in picture

Homework Equations

What is the final answer?

The Attempt at a Solution

I know dw/du = df/dx * dx/du + df/dy * dy/du

& that dx/du = -8sinu & dy/du = -4sinvsinu

Stumped on how to get df/dx and df/dy

It seems odd to me that the question is asking for w_u($\pi/2, 0$) when the partials they give you are evaluated at (0, 0). Unless there's something I'm not thinking of, I would say that there is a typo in the problem.

 

[gunda95]

nope, I asked my prof because I was confused too, but he said that is how it is.

 

[gopher_p]

It seems odd to me that the question is asking for w_u($\pi/2, 0$) when the partials they give you are evaluated at (0, 0). Unless there's something I'm not thinking of, I would say that there is a typo in the problem.

I thought the same until I got my head screwed on straight and realized $w_u(\pi/2, 0)=w_u|_{(u,v)=(\pi/2, 0)}$ and that $x(\pi/2, 0)=y(\pi/2, 0)=0$.

So the more "complete" form of the chain rule that applies here is $$w_u(u,v)=\frac{\partial f}{\partial x}\Big(x(u,v),y(u,v)\Big)\frac{\partial x}{\partial u}(u,v)+\frac{\partial f}{\partial y}\Big(x(u,v),y(u,v)\Big)\frac{\partial y}{\partial u}(u,v)$$ or $$w_u(u,v)=f_x\Big(x(u,v),y(u,v)\Big)x_u(u,v)+f_y\Big(x(u,v),y(u,v)\Big)y_u(u,v)$$

 

[gunda95]

But the final answer is supposed to be a number, wouldn't this give me a answer in coordinates?

 

[gopher_p]

But the final answer is supposed to be a number, wouldn't this give me a answer in coordinates?

What would you get if you plugged in $\pi/2$ for $u$ and $0$ for $v$?