# How do the different concepts of integration fit together?

by V0ODO0CH1LD
Tags: concepts, integration
 P: 259 I'm making this new post in the general math section since I don't know what field of math this question belongs to anymore. So the picture I currently have regarding the abstractions of integration and differentiation from single variable-calculus to multi-variable calculus is that the derivative of a function ##f:\mathbb{R}^n\rightarrow\mathbb{R}^m## abstracts to the Jacobian matrix ##J_f## and that the integral of that derivative abstracts to $$\int_a^bJ_f\cdot{}dx$$ where ##a,b\in\mathbb{R}^n## and ##dx=\langle{}dx_1,dx_2,\ldots,dx_n\rangle##. And this picture makes sense to me because $$\int_a^bJ_f\cdot{}dx=f(b)-f(a)$$ where ##f(a)=\langle{}f_1(a),f_2(a),\ldots,f_m(a)\rangle##. This is all in the case that ##f## is a function such that $$f(x_1,\dots,x_n)= \langle{}f_1(x_1,\dots,x_n),\ldots,f_m(x_1,\dots,x_n)\rangle.$$ Also in this picture, in addition to the usual linear operator nature of ##J_f## it also induces a function ##J_f:\mathbb{R}^n\rightarrow\mathbb{R}^{nm}##. However not every function ##F:\mathbb{R}^n\rightarrow\mathbb{R}^{nm}## is the Jacobian of a function ##f:\mathbb{R}^n\rightarrow\mathbb{R}^m##. But provided a trajectory from the point ##a## to the point ##b## (both in ##\mathbb{R}##) the function ##F## can still be integrated the same way as above, although the fundamental theorem of calculus won't apply (i.e. the value of the integral depends on the trajectory). If this picture is correct (is it?) then my first question is where does the multiple integral fit in it? I realize that there exists an ambiguity in single-variable calculus that the line integral and the multiple integral are basically the same thing, right? But since in vector calculus two points do not specify a closed subset of the domain anymore these two concepts start to differ. Also, I know there are connections between the concepts of line integrals and multiple integrals (via things like Stoke's theorem), but what I am wondering is how to think of them independently so then I can think of these "bridges" between concepts. Unless these "bridges" are the only way to get there :) The other thing I was trying to fit in this picture is the concept of the surface integral. Thanks!