Integrate in multiple dimensions

Question

Somehow the differentiation of functions like $f:\mathbb R^n\rightarrow\mathbb R^m$ seems to be very relevant, whereas the integration of functions like this hardly ever seems to be mentioned. I have the feeling that it simply seems to be obvious to most people, but I'd still like to have some certainty here.

I found some information about multi variable integrals on Wikipedia, where for a function $f:\mathbb R^n\rightarrow\mathbb R, x\mapsto f(x_1,x_2,...,x_n)$ the integral is given by $$\int...\int f(x_1,x_2,...,x_n)dx_1...dx_n$$ so my intuition would state that the integral for the function $$f:\mathbb R^n\rightarrow\mathbb R^m, x\mapsto\begin{pmatrix} f_1(x_1,x_2,...,x_n)\\ f_2(x_1,x_2,...,x_n)\\ ...\\ f_m(x_1,x_2,...,x_n) \end{pmatrix}$$ is given by $$\int\begin{pmatrix} f_1(x_1,x_2,...,x_n)\\ f_2(x_1,x_2,...,x_n)\\ ...\\ f_m(x_1,x_2,...,x_n) \end{pmatrix}dx_1...dx_n=\begin{pmatrix} \int f_1(x_1,x_2,...,x_n)dx_1...dx_n\\ \int f_2(x_1,x_2,...,x_n)dx_1...dx_n\\ ...\\ \int f_m(x_1,x_2,...,x_n)dx_1...dx_n \end{pmatrix}$$ Am I right with this assumption?

And if I am, does the fundamental calculus theorem hold for multiple variables as well? And if so, what derivation do you even look at? In my understanding the equivalent for the derivation in multiple dimensions is the Jacobi-Matrix - but a matrix can hardly generally be the same as the original function.

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Actually I have a precise problem, which is why I'm asking this question. I am supposed to apply the Picard iteration to the following differential equation: $$x'=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}x,\qquad x(0)=\begin{pmatrix} 0\\1 \end{pmatrix}$$ The beginning is quite simple: $$\varphi_0(t)=\begin{pmatrix} 0\\1 \end{pmatrix}$$ But for the next step I need the integral this question is about: $$\varphi_1(t)=\begin{pmatrix} 0\\1 \end{pmatrix}+\int\limits_0^t\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}\varphi_0ds=\begin{pmatrix} 0\\1 \end{pmatrix}+\int\limits_0^t\begin{pmatrix}1\\0\end{pmatrix}ds$$ so I'd calculate it like this: $$\varphi_1(t)=\begin{pmatrix}t\\1\end{pmatrix}$$ Note that my integral is over a function $f:\mathbb R\rightarrow\mathbb R^n$, with $n=2$, which is precisely the part I didn't find an explanation for.

Is this attempt (especially for the Picard-iteration) correct?

Answer 1

Yes, your assumption is correct! The (Riemann) integral of $f$ is simply given by the “element-wise” integrals (if you want to get even more abstract, you can check out the Bochner integral).

The fundamental theorem doesn’t make too much sense here though; the best you can get (if $f$ is “nice”, for instance continuous), is that the corresponding partial derivatives of each coordinate of the integral equals the according coordinate of the original function.

To round it up: Yes! You can (and should ) use the element-wise integral for Picard-Lindelöf, just as you did.

For more information:
Integral of vector-valued function.
Example of Picard-Lindelöf for a system of ODEs