5.7 Change of variables in multiple integrals (Page 5/13)

Calculus volume 3 Page 5 / 13

Using the substitutions $x = v$ and $y = \sqrt{u + v},$ evaluate the integral $\iint_{R} y sin (y^{2} - x) d A$ where $R$ is the region bounded by the lines $y = \sqrt{x}, x = 2, and y = 0 .$

$\frac{1}{2} (sin 2 - 2)$

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Change of variables for triple integrals

Changing variables in triple integrals works in exactly the same way. Cylindrical and spherical coordinate substitutions are special cases of this method, which we demonstrate here.

Suppose that $G$ is a region in $u v w -space$ and is mapped to $D$ in $x y z -space$ ( [link] ) by a one-to-one $C^{1}$ transformation $T (u, v, w) = (x, y, z)$ where $x = g (u, v, w),$ $y = h (u, v, w),$ and $z = k (u, v, w) .$

On the left-hand side of this figure, there is a region G in u v w space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = g(u, v, w), y = h(u, v, w), and z = k(u, v, w). On the right-hand side of this figure there is a region D in xyz space. — A region $G$ in $u v w -space$ mapped to a region $D$ in $x y z -space .$

Then any function $F (x, y, z)$ defined on $D$ can be thought of as another function $H (u, v, w)$ that is defined on $G :$

F (x, y, z) = F (g (u, v, w), h (u, v, w), k (u, v, w)) = H (u, v, w) .

Now we need to define the Jacobian for three variables.

Definition

The Jacobian determinant $J (u, v, w)$ in three variables is defined as follows:

J (u, v, w) = | \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \\ \frac{\partial x}{\partial w} & \frac{\partial y}{\partial w} & \frac{\partial z}{\partial w} \end{matrix} | .

This is also the same as

J (u, v, w) = | \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{matrix} | .

The Jacobian can also be simply denoted as $\frac{\partial (x, y, z)}{\partial (u, v, w)} .$

With the transformations and the Jacobian for three variables, we are ready to establish the theorem that describes change of variables for triple integrals.

Change of variables for triple integrals

Let $T (u, v, w) = (x, y, z)$ where $x = g (u, v, w), y = h (u, v, w),$ and $z = k (u, v, w),$ be a one-to-one $C^{1}$ transformation, with a nonzero Jacobian, that maps the region $G$ in the $u v w -plane$ into the region $D$ in the $x y z -plane .$ As in the two-dimensional case, if $F$ is continuous on $D,$ then

\begin{matrix} \underset{R}{∭} F (x, y, z) d V & = \underset{G}{∭} F (g (u, v, w), h (u, v, w), k (u, v, w)) | \frac{\partial (x, y, z)}{\partial (u, v, w)} | d u d v d w \\ = \underset{G}{∭} H (u, v, w) | J (u, v, w) | d u d v d w . \end{matrix}

Let us now see how changes in triple integrals for cylindrical and spherical coordinates are affected by this theorem. We expect to obtain the same formulas as in Triple Integrals in Cylindrical and Spherical Coordinates .

Obtaining formulas in triple integrals for cylindrical and spherical coordinates

Derive the formula in triple integrals for

cylindrical and
spherical coordinates.

For cylindrical coordinates, the transformation is $T (r, θ, z) = (x, y, z)$ from the Cartesian $r θ z -plane$ to the Cartesian $x y z -plane$ ( [link] ). Here $x = r cos θ,$ $y = r sin θ,$ and $z = z .$ The Jacobian for the transformation is

$\begin{matrix} J (r, θ, z) & = \frac{\partial (x, y, z)}{\partial (r, θ, z)} = | \begin{array}{l} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial θ} & \frac{\partial x}{\partial z} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial θ} & \frac{\partial y}{\partial z} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial θ} & \frac{\partial z}{\partial z} \end{array} | \\ = | \begin{array}{l} cos θ & - r sin θ & 0 \\ sin θ & r cos θ & 0 \\ 0 & 0 & 1 \end{array} | = r {cos}^{2} θ + r {sin}^{2} θ = r ({cos}^{2} θ + {sin}^{2} θ) = r . \end{matrix}$

We know that $r \geq 0,$ so $| J (r, θ, z) | = r .$ Then the triple integral is

$\underset{D}{∭} f (x, y, z) d V = \underset{G}{∭} f (r cos θ, r sin θ, z) r d r d θ d z .$

The transformation from rectangular coordinates to cylindrical coordinates can be treated as a change of variables from region $G$ in $r θ z -space$ to region $D$ in $x y z -space .$
For spherical coordinates, the transformation is $T (ρ, θ, φ) = (x, y, z)$ from the Cartesian $p θ φ -plane$ to the Cartesian $x y z -plane$ ( [link] ). Here $x = ρ sin φ cos θ,$ $y = ρ sin φ sin θ,$ and $z = ρ cos φ .$ The Jacobian for the transformation is

$J (ρ, θ, φ) = \frac{\partial (x, y, z)}{\partial (ρ, θ, φ)} = | \begin{array}{l} \frac{\partial x}{\partial ρ} & \frac{\partial x}{\partial θ} & \frac{\partial x}{\partial φ} \\ \frac{\partial y}{\partial ρ} & \frac{\partial y}{\partial θ} & \frac{\partial y}{\partial φ} \\ \frac{\partial z}{\partial ρ} & \frac{\partial z}{\partial θ} & \frac{\partial z}{\partial φ} \end{array} | = | \begin{array}{l} sin φ cos θ & - ρ sin φ sin θ & ρ cos φ cos θ \\ sin φ sin θ & - ρ sin φ cos θ & ρ cos φ sin θ \\ cos θ & 0 & - ρ sin φ \end{array} | .$

Expanding the determinant with respect to the third row:

$\begin{matrix} = cos φ | \begin{array}{l} - ρ sin φ sin θ & ρ cos φ cos θ \\ ρ sin φ sin θ & ρ cos φ sin θ \end{array} | - ρ sin φ | \begin{array}{l} sin φ cos θ & - ρ sin φ sin θ \\ sin φ sin θ & ρ sin φ cos θ \end{array} | \\ = cos φ (− ρ^{2} sin φ cos φ {sin}^{2} θ - ρ^{2} sin φ cos φ {cos}^{2} θ) \\ - ρ sin φ (ρ {sin}^{2} φ {cos}^{2} θ + ρ {sin}^{2} φ {sin}^{2} θ) \\ = − ρ^{2} sin φ {cos}^{2} φ ({sin}^{2} θ + {cos}^{2} θ) - ρ^{2} sin φ {sin}^{2} φ ({sin}^{2} θ + {cos}^{2} θ) \\ = − ρ^{2} sin φ {cos}^{2} φ - ρ^{2} sin φ {sin}^{2} φ \\ = − ρ^{2} sin φ ({cos}^{2} φ + {sin}^{2} φ) = − ρ^{2} sin φ . \end{matrix}$

Since $0 \leq φ \leq π,$ we must have $sin φ \geq 0 .$ Thus $| J (ρ, θ, φ) | = | − ρ^{2} sin φ | = ρ^{2} sin φ .$

The transformation from rectangular coordinates to spherical coordinates can be treated as a change of variables from region $G$ in $ρ θ φ -space$ to region $D$ in $x y z -space .$

Then the triple integral becomes

$\underset{D}{∭} f (x, y, z) d V = \underset{G}{∭} f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ^{2} sin φ d ρ d φ d θ .$

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Source: OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2

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