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True-False Determine whether the statement is true or false. Explain your answer.If $f(x, y)$ and $g(x, y)$ are differentiable functions defined on the $x y$ -plane, and if $f_{y}(x, y)=g_{x}(x, y)$ for all $(x, y)$, then there exists a function $\phi(x, y)$ such that $\phi_{x}(x, y)=f(x, y)$ and $\phi_{y}(x, y)=g(x, y)$.

Key Concepts

Integrability Condition

The integrability condition, which requires that the mixed partial derivatives f_y and g_x be equal, is necessary for a differential form to be exact. This condition ensures that the differential of ? is independent of the path taken, making it possible to define ? globally by integrating the given functions.

Simply Connected Domain

A simply connected domain is a region in which every closed loop can be continuously contracted to a point. This property is important because when a differential form satisfies the integrability condition, it guarantees the existence of a potential function ? on such domains. The entire xy-plane is simply connected, which is why the condition f_y = g_x is sufficient here.

Exact Differential

An exact differential in two variables means that there is a function ?(x, y) whose total differential, d?, equals f(x, y) dx + g(x, y) dy. The existence of such a function implies that f and g are, in fact, the partial derivatives of ? with respect to x and y, respectively.

Clairaut's Theorem (Mixed Partial Derivatives)

Clairaut's Theorem states that the mixed partial derivatives of a function are equal if they are continuous. In this context, since f and g are differentiable (implying continuity of the relevant partial derivatives), the condition f_y = g_x is not only necessary but also sufficient for the existence of the potential function ?.

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