Hello,
Please try to solve this question.
Thanks.
The question is asking in other words, does there exists a function $\displaystyle z=f(x,y)$.
So that, $\displaystyle \nabla f(x,y) = e^x \sin y \bold{i} + e^x \cos y \bold{j}$
We need that,
$\displaystyle \frac{\partial f}{\partial x} = e^x \sin y$
$\displaystyle \frac{\partial f}{\partial y} = e^x \cos y$
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:
$\displaystyle f = e^x \sin y + g(y)$
But that means if we differenciate to respect to y:
$\displaystyle \frac{\partial f}{\partial y} = e^x \cos y + g'(y) = e^x \cos y $
So $\displaystyle g'(y) = C$ will work for any constant. We can just choose $\displaystyle C=0$.
Thus, $\displaystyle f(x,y) = e^x \sin y$ is a function which works.