- #1
Abdullah Almosalami
- 49
- 15
Is there such a thing as an antiderivative of a multivariable function? I haven't put too much thought into this yet but I wanted to ask anyways. Sticking for now just to two variables, I was observing that double integrals are always definite integrals, whereas in the single-variable case, we have both definite and indefinite integrals. I suppose I am asking given a function ##f(x,y)##, is there any meaningful relationship between a function ##f(x,y)## and some other function ##F(x,y)## such that ## \frac {\partial^2 F} {\partial x \partial y} = f(x,y) ##? Moreover, what I am getting at after that is is there some kind of analogous Fundamental Theorem of Calculus for two-variable functions that goes something like ##\int_{y=c}^d \int_{x=a}^b f(x,y) \, dx \, dy = F(P_1) - F(P_2)## where ##P_1## and ##P_2## are some points in the domain of ##F(x,y)##? I'm not sure if this is down the line my Calculus textbook, so it might literally be in the next section.