Find a function f(x,y) such that the gradient is

Question

Problem: Find a function $f(x,y)$ such that $ \nabla f = <y,x>$

My work:

$\dfrac {\partial f}{\partial x} = y$

$\dfrac {\partial f}{\partial y} = x$

$f(x,y) = \displaystyle\int_{ }^{ } \dfrac{\partial f}{\partial x} dx + \int_{ }^{ } \dfrac {\partial f}{\partial y} dy = yx + xy = 2xy$

$\dfrac {\partial (2xy)}{\partial x} = 2x$

$\dfrac {\partial (2xy)}{\partial y} = 2y$

Then shouldn't the function be $xy$?

Answer 1

$$df={(df/dx)_y}dx+{(df/dy)_x}dy=ydx+xdy=d(xy)$$ so $$f=xy$$.

Answer 2

Your method of constructing the function isn't really right. You can proceed as follows:

Since $\partial f / \partial x = y$ then

$$f(x,y) = \int \frac{\partial f}{\partial x} \, dx = xy + h(y).$$

Differentiating this with respect to $y$ gives

$$\frac{\partial f}{\partial y} = x + h'(y) = x, \text{ therefore } h'(y) = 0 \text{ and } h(y) = C.$$

Hence the family of functions is $f(x,y) = xy +C$.