3.3 Expressing $f$ as $\nabla \varphi$

We begin with a remark on higher derivatives.

You’ve seen higher partial derivatives $\frac{{\partial }^{2}f}{\partial x^2}=f_{xx}$, $f_{xy}$, $f_{yx}$ and $f_{yy}$ in MATH102. Recall the following important fact:

Fact: If $f_{x\mathit{}y}$ and $f_{y\mathit{}x}$ exist and are continuous, then they are equal.

Since (assuming continuity) ${\varphi }_{xy}={\varphi }_{yx}$, there is an obvious necessary condition for such a $\varphi$ to exist: we must have ${\displaystyle \frac{\partial f_1}{\partial y}}={\displaystyle \frac{\partial f_2}{\partial x}}$. Conversely, one can show that if this condition holds, then there is indeed such a $\varphi$.

The required condition is satisfied, since

To find $\varphi$: we require ${\varphi }_x=2x+3y$. Integration with respect to $x$ (with $y$ counting as constant) gives

From the equalities ${\varphi }_{yz}={\varphi }_{zy}$ (etc.), this can only happen if

and again one can show that these conditions are sufficient to ensure the existence of $\varphi$.

The conditions are satisfied: