Editing Partial derivative (section)

==Antiderivative analogue==
There is a concept for partial derivatives that is analogous to [[antiderivative]]s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of 

<math display="block">\frac{\partial z}{\partial x} = 2x+y.</math>

The so-called partial integral can be taken with respect to {{mvar|x}} (treating {{mvar|y}} as constant, in a similar manner to partial differentiation):

<math display="block">z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y).</math>

Here, the [[constant of integration]] is no longer a constant, but instead a function of all the variables of the original function except {{mvar|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 {{mvar|x}} 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 {{nowrap|<math>x^2 + xy + g(y)</math>,}} where {{mvar|g}} is any one-argument function, represents the entire set of functions in variables {{math|''x'', ''y''}} that could have produced the {{mvar|x}}-partial derivative {{nowrap|<math>2x + y</math>.}}

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. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function. In other words, not every vector field is [[Conservative vector field|conservative]].