Hey, id love some help in doing this questions. Cheers
1. Integrate $\displaystyle \frac{\partial \phi}{\partial x} = 4y~cos(4xy)$ over x. This will be a function f(x, y, z) plus some function g(y, z).
For example, let $\displaystyle \frac{\partial \phi}{\partial x} = 3x + 4xy \implies \phi = \frac{3}{2}x^2 + 2x^2y + g(y, z)$
The function g is a constant as far as the integration is concerned, so we have g(y, z) as an arbitrary "constant."
Problems 2 and 3 use the same idea. See if you can get those for now.
-Dan