This is a sub-page of our page on Integration (of functions of several real variables)

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The sub-pages of this page are:

• Separating the variables
• Different Coordinate Systems

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Related KMR-pages:

• …

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Other relevant sources of information:

• Multiple integral
• Change of variables
• Change of variables and the Jacobian, by Serpentine Integral on YouTube.
• Integral
• Determinant

Anchors into the text below:

• The dirty little secret of the multiple Riemann integral

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Substitution in integrals

In one variable, we are mapping from a line to a line. A line segment (such as dx) can be given an “intrinsic” direction (= a direction that does not depend on any coordinate representation). In “ordinary” (= commutative) algebra directional ordering is impossible for higher-dimensional manifolds (such as planes or volumes). This is the reason behind the very important difference (formulated below as an “important caveat”) between the integration-variable-substitution formula in the case of one variable and the case of several variables.

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One variable:

\, \begin{matrix} \mathbf{R} & \xleftarrow{{{\int} \atop {D}}} & \mathbf{R} & \xleftarrow{f} & \mathbf{R}^1 & \xleftarrow{x} & \mathbf{R}^1 \\ & & & & & & & \\ {{\int} \atop {D}} f(x) dx & \longleftarrow & f(x) & \longleftarrow & x & & \\ & & & & & & & \\ & & & & D & & \\ & & & & & & & \\ {{\int} \atop {x^{-1}(D)}} f(x(u)) \det x'(u) \, du & \longleftarrow & f(x(u)) & \longleftarrow & x(u) & \longleftarrow & u \\ & & & & & & & \\ & & & & & & {x^{-1}(D)} \\ \end{matrix} \,

These two integrals express the same result, although they differ in the way this result is computed. Hence we get the so-called “substitution formula” for changing the running variable of an integral:

\, {{\int} \atop {D}} f(x) \, dx \, = \, {{\int} \atop {x^{-1}(D)}} f(x(u)) \det x'(u) \, du \, .

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The dirty little secret of the multiple Riemann integral

In two dimensions, \, dx \, = \, dx_1 \, dx_2 \, represents an infinitesimal un-directed surface (= 2-volume) element , and in three dimensions, \, dx \, = \, dx_1 \, dx_2 \, dx_3 \, represents an infinitesimal un-directed 3-volume element, and so on. Moreover, the derivative corresponds to the so-called Jacobian matrix:

\, x'(u) \, = \, \begin{pmatrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} & \cdots & \frac{\partial x_1}{\partial u_n} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} & \cdots & \frac{\partial x_2}{\partial u_n} \\ & & \ddots & \\ \frac{\partial x_n}{\partial u_1} & \frac{\partial x_n}{\partial u_2} & \cdots & \frac{\partial x_n}{\partial u_n} \end{pmatrix} \, ,

and the sign of the derivative corresponds to the sign of its determinant \, \det x'(u) .

\, \begin{matrix} \mathbf{R} & \xleftarrow{{{\int} \atop {D}}} & \mathbf{R} & \xleftarrow{f} & \mathbf{R}^n & \xleftarrow{x} & \mathbf{R}^n \\ & & & & & & & \\ {{\int} \atop {D}} f(x) \, dx & \longleftarrow & f(x) & \longleftarrow & x & & \\ & & & & & & & \\ & & & & D & & \\ & & & & & & & \\ {{\int} \atop {x^{-1}(D)}} f(x(u)) \, \det x'(u) \, du & \longleftarrow & f(x(u)) & \longleftarrow & x(u) & \longleftarrow & u \\ & & & & & & & \\ & & & & & & {x^{-1}(D)} \\ \end{matrix} \,

Undirected volume element:

\, {{\int} \atop {D}} f(x) \, dx \, = \, {{\int} \atop {x^{-1}(D)}} f(x(u)) \det x'(u) \, du \, .

\, \textcolor{red}{Important \; caveat \; for \; Riemann \; integration} \, : This formula is only valid
if the determinant \, \det x'(u) \, of the Jacobian matrix \, x'(u) \, does not change its sign
within the domain of integration \, D .

Example (in two variables):

\, {{\int} \atop {D}} f(x) \, dx_1 \, dx_2 = {{\int} \atop {x^{-1}(D)}} f(x(u)) \, \det \begin{pmatrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \end{pmatrix} du_1 \, du_2 \, ,

if the determinant does not change its sign within the domain of integration \, D .

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Introducing directed volume elements:

By making use of the so-called wedge product (or exterior product), one can introduce directed volume elements. In the case of such order-sensitive infinitesimal volume elements we will write

\, dx \, = \, dx_1 \wedge dx_2 \, , \, dx \, = \, dx_1 \wedge dx_2 \wedge dx_3 \, , and so on.

The order-sensitive (= parity-sensitive) integral obtained in this way has several advantages in relation to the standard (order-insensitive) Riemann integral. The most important advantage is the fact that one does not have to know in advance where the Jacobian matrix \, x'(u) \, changes the sign of its determinant.

Example (in two variables):

\, {{\int} \atop {D}} f(x) \, dx_1 \wedge dx_2 \, = \, {{\int} \atop {x^{-1}(D)}} f(x(u)) \, \det \begin{pmatrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \end{pmatrix} du_1 \wedge du_2 \, .

In fact, this type of integral, which is used in so-called Geometric Calculus, transforms exactly the same way as a one-dimensional integral under substitution. Therefore the one-dimensional integration-variable-substitution formula becomes valid in several variables without any restrictions on the sign of the determinant.

Hence the formula \, {{\int} \atop {D}} f(x) \, dx \, = \, {{\int} \atop {x^{-1}(D)}} f(x(u)) \, \det x'(u) \, du \,
is valid across the entire domain of integration – whether the Jacobian determinant is changing its sign or not.

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Representation: \, [ \, p_{resentant} \, ]_{B_{ackground}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{ackground}}

\, [ \, c_{hange} \, o_{f} \, v_{ariable} \, ]_{G_{eometric} \, I_{ntegration}} \, \mapsto \, \, \left< \, {{\int} \atop {D}} f(x) \, dx \, = \, {{\int} \atop {x^{-1}(D)}} f(x(u)) \, \det x'(u) \, du \, \right>_{G_{eometric} \, I_{ntegration}}

without any restrictions on the sign of the Jacobian determinant \, \det x'(u) .

This parity-sensitive “geometric integral” (also called a “clifford integral”) has important applications, for example in chemistry, where it is often advantageous to integrate over regions where the Jacobian determinant changes its sign.

Here is an illustration of the restrictions involved with having to know
where the Jacobian determinant changes its sign:

If the same order-insensitive situation as for Riemann integrals of several real variables were to apply for Riemann integrals of one real variable, then, in order to integrate a smooth function \, f : \, \mathbb{R} \rightarrow \mathbb{R} \, , one would have to know in advance where the derivative of \, f \, changes its sign, because this derivative is the one-dimensional version of the Jacobian matrix, and therefore, the sign of the Jacobian determinant is identical to the sign of this derivative.

Hence, under such order-insensitive circumstances, it would be impossible to integrate a function over an interval where its derivative changes sign. This restriction would introduce an absurd limitation of the power and applicability of one-dimensional Riemann integrals.

/////// Elaborate on:

• Show how the chain rule interacts with this.
•The Jacobian matrix (= the local linearization), and
• The Jacobian determinant (= the local (directed) volume ratio).

/////// Formula repository for this page:

x = x(u):

\, x(\textcolor{blue} {p} + \textcolor{red} {\Delta u}) = x(\textcolor{blue}{p}) + \textcolor{blue} {x'(\textcolor{black} {u})_p} \textcolor{red} {\Delta u} + o(\textcolor{red} {\Delta u}) \, .

f(x) = f(x(u)):

\, f(x(\textcolor{blue} {p} + \textcolor{red} {\Delta u})) = f(x(\textcolor{blue} {p})) + \textcolor{blue} {f'(\textcolor{black} {x})_{x(p)} \textcolor{blue} {x'(\textcolor{black} {u})_p}} \textcolor{red} {\Delta u} + o(\textcolor{red} {\Delta u}) \, .

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