       Multi-variable Integration

• To: mathgroup at smc.vnet.net
• Subject: [mg79666] Multi-variable Integration
• From: gravmath at yahoo.com
• Date: Wed, 1 Aug 2007 05:07:34 -0400 (EDT)

```Suppose one defines two expressions in Mathematica:

Q = f[x,y]
P = f[x,y] + g[y]

and then differentiates them wrt the variable 'x' as follows:

dQ = D[Q,x]
dP = D[P,x]

Subsequent use of the Integrate command wrt the variable
'x' (Integrate[dQ,x] and Integrate[dP,x]) yields, in both cases,
f[x,y].

I find this behavior understandable from a systems point of view but
mathematically in both cases the answer should be f[x,y] +
arbitraryfunc[y], where obviously further input (as in the original
definitions of P & Q) is needed to determine arbitraryfunc[y].  Is
there a way to get Mathematica to recognize that there are two
variables in the problem and to produce the arbitrary function of the
variable 'y'?

I'm guessing that my specification of f[x,y] is not quite sufficient
to do this, even though it is sufficient when differentiating.  That
is to say that dQ and dP are rendered in Mathematica as f^(1,0)[x,y],
which clearly indicates that Mathematica understands that there are
two independent variables in the expression.

Any help would be appreciated, even if it to point me to previous
posts (I found no germane ones myself).