Integration on Rn

Schuller GR.
Throughout this note, we assume that all integrals we write down are well-defined (i.e., they exist in the appropriate sense, such as Riemann or Lebesgue).
The simplest case is that of a function $F : R \to R$ , where we simply have

\int_{(a, b)} F := \int_{a}^{b} d x F (x),

where the right-hand side integral is some known integration operation (e.g., Riemann integrals, Lebesgue integrals).

Next we can consider $F : R^{d} \to R$ . If we are to do this over a box-shaped domain, $(a, b) \times \dots \times (u, v) \subset R^{d}$ , the integral is simply

\int_{(a, b) \times \dots \times (u, v)} d^{d} x F (x) := \int_{a}^{b} d x^{1} \dots \int_{u}^{v} d x^{d} F (x^{1}, \dots, x^{d}) .

We can then extend this to general domains (i.e., not necessarily box-shaped) $G \subset R^{d}$ by introducing an indicator function $μ_{G} : R^{d} \to R$ given by

μ_{G} (x) = {\begin{cases} 1 & if x \in G \\ 0 & otherwise . \end{cases}

We then define the integral

\int_{G} d^{d} x F (x) := \int_{- \infty}^{\infty} d x^{1} \dots \int_{- \infty}^{\infty} d x^{d} μ_{G} (x) F (x) .

It represents the generalized volume of a container with base $U$ and whose variable height is given by $F$ .

\usepackage{tikz-3dplot}
\begin{document} 

\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords]
% Define the coordinates of the base (subset U)
\coordinate (A) at (-2,1,0);
\coordinate (B) at (-1,2,0);
\coordinate (C) at (1,2,0);
\coordinate (D) at (2,1,0);
\coordinate (E) at (1,-1,0);
\coordinate (F) at (-1,-1.5,0);

% Draw the base (irregular subset U)
\draw[fill=blue!20, opacity=0.7] 
    plot [smooth cycle] 
    coordinates {(A) (B) (C) (D) (E) (F)} node[below right]{$U$};

% Define the heights of the surface points
\coordinate (A') at (-2,1,2.8);
\coordinate (B') at (-1,2,2.5);
\coordinate (C') at (1,2,3.2);
\coordinate (D') at (2,1,2.8);
\coordinate (E') at (1,-1,2.5);
\coordinate (F') at (-1,-1.5,2.7);

% Draw the surface defined by F(x, y)
\draw[fill=red!20, opacity=0.7] 
    plot [smooth cycle] 
    coordinates {(A') (B') (C') (D') (E') (F')};

% Draw dashed lines connecting base to surface (prism walls)
\foreach \base/\top in {A/A', B/B', C/C', D/D', E/E', F/F'} {
    \draw[dashed] (\base) -- (\top);
}

% Axis
\draw[thick,->] (-3,0,0) -- (3,0,0) node[anchor=north east]{$x$};
\draw[thick,->] (0,-3,0) -- (0,3,0) node[anchor=north west]{$y$};
\draw[thick,->] (0,0,0) -- (0,0,4) node[anchor=south]{$z$};

\end{tikzpicture} 
\end{document}

We now need to ask how this definition changes under a change of variable (which will correspond to a change of chart in the lifted notion).

Theorem: Let $ϕ : ϕ^{- 1} (G) \to G$ denote the change of variable map with $G \subset R^{d}, ϕ^{- 1} (G) \subset R^{d}$ . Then if the integral of $F : G \to R$ is defined as above, we have

\int_{G} d^{d} x F (x) = \int_{ϕ^{- 1} (G)} d^{d} y | det (\frac{\partial ϕ^{a}}{\partial y^{b}}) (y) | (F \circ ϕ) (y),

where $| det (\frac{\partial ϕ^{a}}{\partial y^{b}}) (y) |$ is the Jacobian of $ϕ$ . Sometimes the Jacobian is defined without the absolute value part, but here we shall use the whole thing. $◼$

Integration of forms

The change of variables theorem implies that we can interpret integration as if we are actually integrating $n$ -forms. That is, if we have in $R^{n}$ a set $U$ and a function $f$ , both could be reinterpreted in a new coordinate system, given by $ϕ$ , as $\tilde{G} = ϕ (G)$ and $\tilde{f} = f \circ ϕ^{- 1}$ . If we think of $f$ not like a function but like an $n$ -form (by using the standard volume form in $R^{n}$ ), then after the change $ω = f d x$ will become in the $n$ -form $\tilde{ω} = J a c (ϕ) \cdot f \circ ϕ^{- 1} d y$ and therefore

\int_{U} ω = \int_{\tilde{U}} \tilde{ω}

From here, we can go to integration on manifolds.