Is there such a thing as an indefinite multiple integral?

[swampwiz]

A calculus book I am looking at ("Early Transcendentals") does not seem to mention it, but it seems like it is possible, with the constants of integration being functions that have no component in variable being used for the integration.

Any insight? Articles?

 

[AlephZero]

If you regard a multiple integral as the solution to a partial differential equation, an "indefinte multiple integral" would correspond to a general solution with no boundary conditions specified, and such solutions usually contain arbitrary functions rather than arbitary constants.

So in that sense, the answer to your question is yes.

But this idea probably isn't so useful as indefinite integrals of a single variable. For ordinary differential equations, finding the general solujtion as an indefinite integral and using the boundary conditions to fix the undetermined constant(s) is a fairly standard solution technique. For partial differential equations, the "general" solution is often so general that it can be written many different ways with apparently different types of arbitrary functions, so to get to a particular solution you have to choose the appropriate form of "general" solution to work with, and that doesn't just fall out of the multiple integral.

As a simple example, take the linear wave equation in 3D space. The "general" solutions can be plane waves, cylindrical waves, spherical waves, etc, etc ... and if the boundary conditions are something to do with a sphere, trying to find the particular solution starting form a general solution expressed as arbitrary functions representing plane waves is going to be somewhere between very difficult and impossible.

 

[TylerH]

That's an interesting question. Wikipedia says: "Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral." So I guess not.

http://en.wikipedia.org/wiki/Multiple_integral#Introduction

 

[JJacquelin]

The subject of multiple integrals is not new. A famous formula from Cauchy was published in 1823.
For example, see Reference [4] in the paper "La dérivation fractionnaire", Section 3 :
http://www.scribd.com/JJacquelin/documents
In attachment, an oversimplified insight of the background.