# Thread: evaluate z

1. ## evaluate z

Hello,
Please try to solve this question.
Thanks.

2. The question is asking in other words, does there exists a function $z=f(x,y)$.

So that, $\nabla f(x,y) = e^x \sin y \bold{i} + e^x \cos y \bold{j}$

We need that,
$\frac{\partial f}{\partial x} = e^x \sin y$

$\frac{\partial f}{\partial y} = e^x \cos y$

To see if this if this is possible we can use the "cross-partials test". Or just assume that it is possible and see if we arrive at a result, which is what I am going to do.

The first partial differencial equation solves to:
$f = e^x \sin y + g(y)$

But that means if we differenciate to respect to y:
$\frac{\partial f}{\partial y} = e^x \cos y + g'(y) = e^x \cos y$

So $g'(y) = C$ will work for any constant. We can just choose $C=0$.

Thus, $f(x,y) = e^x \sin y$ is a function which works.

3. ## another way

there is another way to solve the problem

4. Originally Posted by curvature
there is another way to solve the problem
I think you did the wrong problem.

5. Originally Posted by ThePerfectHacker
I think you did the wrong problem.
Sorry I uploaded the wrong image. It has been replaced by the correct one.