find f if grad f = 3xy i + x^3 j
IF there exist a function f having that gradient, then we must have $\displaystyle \frac{\partial f}{\partial x}= 3xy$ and $\displaystyle \frac{\partial f}{\partial y}= x^3$. If we differentiate the first a second time, but with respect to y, we have $\displaystyle \frac{\partial^2 f}{\partial y\partial x}= 3x$. If we differentiate the second a second time, but with respect to x, we have $\displaystyle \frac{\partial^2 f}{\partial x\partial y}= 3x^2$. As long as all derivatives are continuous, as they are here, the "mixed" second derivatives must be the same. Since this is not true, there is NO function, f, having 3xyi+ x^3j as its gradient.