find f if grad f = 3xy i + x^3 j
IF there exist a function f having that gradient, then we must have and . If we differentiate the first a second time, but with respect to y, we have . If we differentiate the second a second time, but with respect to x, we have . 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.