Section 6.5 Comments on More Variables
There is no real difficulty about generalising the theory in this section to more than three variables. The integral is defined subdividing a domain in \(\mathbb R^N\) into small “boxes”, that is, Cartesian products of \(N\) rather than three small intervals, and then doing the same kind of limiting process. Everything, including the transformation formula in Theorem 6.5 remains valid. Also the proof of the transformation formula does not require new ideas, and relies on the facts on \(N\)-dimensional volumes in Remark 1.31. Also spherical coordinates as well as various cylindrical coordinates have a generalisation to \(N\) variables, \(N>3\text{.}\)
Who wants to know about “volumes” of sets in four, five higher “dimensions”, sets which we cannot even draw? To answer this question recall that the interpretation of the double integral as the volume of a body is only one of many other possible interpretation! We saw in Remark 5.9 that if \(\varrho(\vect x)\) is the mass density of a plate occupying \(D\) then
\begin{equation*}
\int_D\varrho(\vect x)\,d\vect x
\end{equation*}
is the total mass of the plate. This is a problem with a natural extension to a body of mass in space. If \(\varrho(x,y,z)\) is the mass density of a body in space at \((x,y,z)\) then computing the total mass leads to the integral of a function, \(\varrho\text{,}\) of three variables, which we called atriple integral. Hence we have one model problem for integrals of functions of three variables. The problem however does not make sense to motivate integrals of functions of four or more variables.
We will give one (very simplified) example from physics. A radio transmitter produces electro-magnetic waves which can be received by a radio, television or a mobile phone. The antenna of one of these devices absorbs the energy of the electro-magnetic field, amplifies the signal and turns it into something we can understand. We want to compute how much energy an antenna absorbs from the electro-magnetic field in a given time interval. If \(\vect E(x,y,z,t)\) is the electric and \(\vect B(x,y,z,t)\) the magnetic field at the point \((x,y,z)\) at time \(t\text{,}\) then it is shown in physics that its `energy density’ (energy per volume) at \((x,y,z)\) at time \(t\) is given by
\begin{equation*}
\frac{1}{2}\Bigl(\|\vect E(x,y,z,t)\|^2+\|\vect
B(x,y,z,t)\|^2\Bigr)
\end{equation*}
Assume that the antenna of the receiving device occupies the region \(R\subset\mathbb R^3\text{.}\) To compute how much energy the antenna absorbs in the time interval \([a,b]\) we first compute the energy of the electro-magnetic field in \(R\) at every instance \(t\in[a,b]\text{.}\) It is the `sum’ of the density over the region \(D\text{,}\) represented by the triple integral
\begin{equation*}
\iiint_D\frac{1}{2}\Bigl(\|\vect E(x,y,z,t)\|^2+\|\vect B(x,y,z,t)\|^2\Bigr)\,dx\,dy\,dz\text{.}
\end{equation*}
To get the energy absorbed in the time interval \([a,b]\) we have to take the `sum’ over \([a,b]\text{,}\) that is, integrate over \([a,b]\text{.}\) Hence the total energy absorbed is
\begin{equation*}
\iiiint_{[a,b]\times D}\frac{1}{2}
\Bigl(\|\vect E(x,y,z,t)\|^2+\|\vect B(x,y,z,t)\|^2\Bigr)\,dx\,dy\,dz\,dt\text{,}
\end{equation*}
which is the integral of a function of four variables. Other problems require taking integrals over functions of even more variables, but we hope the above provides enough motivation for the theory developed.
