Partial Derivative - Antiderivative Analogue

Antiderivative Analogue

There is a concept for partial derivatives that is analogous to antiderivatives for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of . The "partial" integral can be taken with respect to x (treating y as constant, in a similar manner to partial derivation):

Here, the "constant" of integration is no longer a constant, but instead a function of all the variables of the original function except x. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the "constant" represent an unknown function of all the other variables.

Thus the set of functions, where g is any one-argument function, represents the entire set of functions in variables x,y that could have produced the x-partial derivative 2x+y.

If all the partial derivatives of a function are known (for example, with the gradient), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant.