$S$

$\mathrm{d\σ}$

Explicit

$z\=z\left(x\,y\right)$

$\sqrt{1\+z_x^2\+z_y^2} \mathrm{dA}$

Implicit

$f\left(x\,y\,z\right)\=0\Rightarrow z\=z\left(x\,y\right)$

$\frac{∥\nabla f∥}{\|f_z\|} \mathrm{dA}$ 

Parametric

$\{\begin{array}{c}x\=x\left(u\,v\right)\\ y\=y\left(u\,v\right)\\ z\=z\left(u\,v\right)\end{array}$ 

$\sqrt{J_1^2\+J_2^2\+J_3^2}\mathrm{dA}$, where

$J_1\=\frac{\partial \left(y\,z\right)}{\partial \left(u\,v\right)}$ = $\left|\begin{array}{cc}{y}_y& y_v\\ z_u& z_v\end{array}\right|$ = $y_u{z}_v-y_v{z}_u$ 

$J_2\=\frac{\partial \left(z\,x\right)}{\partial \left(u\,v\right)}$ = $\left|\begin{array}{cc}{z}_u& z_v\\ x_u& x_v\end{array}\right|$ = $z_u{x}_v-z_v{x}_u$ 

$J_3\=\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}$ = $\left|\begin{array}{cc}{x}_u& x_v\\ y_u& y_v\end{array}\right|$ = $x_u{y}_v-x_v{u}_u$ 

Table 9.6.1   Forms of the surface-area element $\mathrm{d\σ}$ 

$S$

$\int {\int }_{S}g \mathrm{d\σ}$

Explicit

$z\=z\left(x\,y\right)$

$\int {\int }_{S}g\left(x\,y\,z\left(x\,y\right)\right) \mathrm{d\σ}$ $$

Implicit

$f\left(x\,y\,z\right)\=0\Rightarrow z\=z\left(x\,y\right)$

Parametric

$\{\begin{array}{c}x\=x\left(u\,v\right)\\ y\=y\left(u\,v\right)\\ z\=z\left(u\,v\right)\end{array}$ 

$\int {\int }_{S}g\left(x\left(u\,v\right)\,y\left(u\,v\right)\,z\left(u\,v\right)\right) \mathrm{d\σ}$

Table 9.6.2   Surface integral of the scalar function $g\left(x\,y\,z\right)$ 

Surface Integration

Scalar Function

Vector Field

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻

Surface Integration

Flux≻3-D

Over a Box

Through a Box

Over a Parametrically Defined Surface

Through a Parametric Surface

Over a Sphere

Through a Sphere

Surface Defined over a 2-D Region

Through a Surface Defined over a Planar Region

Surface Defined over a Disk

Through a Surface Defined over a Disk

Surface Defined over a Rectangle

Surface Defined over a Triangle

Through a Surface Defined over a Triangle

Surface Defined over an Ellipse

Through a Surface Defined over an Ellipse

Table 9.6.3   Tasks templates for surface integration

1

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Sphere}\left(⟨0\,0\,0⟩\,a\right)\,\mathrm{output}\=\mathrm{integral}\right)$

2

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Box}\left(1..2\,3..4\,5..6\right)\,\mathrm{output}\=\mathrm{integral}\right)$

3

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,x\=0..\mathrm{\π}\/4\,y\= \sin \left(x\right)..\cos \left(x\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

4

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Region}\left(0..\mathrm{\π}\/4\, \sin \left(x\right)..\cos \left(x\right)\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

5

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,x\=-1..1\,y\=-2..2\right)\,\mathrm{output}\=\mathrm{integral}\right)$

6

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Rectangle}\left(-1..1\,-2..2\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

7

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨0\,0⟩\,a\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

8

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Sector}\left(\mathrm{Circle}\left(⟨0\,0⟩\,a\,\left[r\,\mathrm{\theta }\right]\right)\,0\,\mathrm{\π}\/4\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

9

$\mathrm{simplify}\left(\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+2 y^2\=1\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

10

$\mathrm{simplify}\left(\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(⟨0\,0⟩\,2\,1\,\left[r\,\mathrm{\theta }\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

11

$\mathrm{simplify}\left(\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Sector}\left(\mathrm{Ellipse}\left(x^2\+2 y^2\=1\,\left[r\,\mathrm{\theta }\right]\right)\,0\,\mathrm{\π}\/4\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

12

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨1\,2⟩\,⟨5\,3⟩\,⟨4\,7⟩\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

Table 9.6.4   Examples of valid syntax for the SurfaceInt command

Table 9.6.5   Comments on Table 9.6.4

1

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Box}\left(1..2\,3..4\,5..6\right)\,\mathrm{output}\=\mathrm{integral}\right)$

2

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Sphere}\left(⟨0\,0\,0⟩\,a\right)\,\mathrm{output}\=\mathrm{integral}\right)$

3

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,x\=0..1\,y\=0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

4

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,x\=0..\mathrm{\π}\/4\,y\=\sin \left(x\right)..\cos \left(x\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

5

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,0..1\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

6

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Rectangle}\left(0..1\,0..1\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

7

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Region}\left(0..\mathrm{\π}\/4\,\sin \left(x\right)..\cos \left(x\right)\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

8

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨0\,0⟩\,a\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

9

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Sector}\left(\mathrm{Circle}\left(⟨0\,0⟩\,a\,\left[r\,\mathrm{\theta }\right]\right)\,0\,\mathrm{\π}\/4\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

10

$\mathrm{simplify}\left(\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+2 y^2\=1\,\left[r\,\mathrm{\theta }\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

11

$\mathrm{simplify}\left(\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Sector}\left(\mathrm{Ellipse}\left(x^2\+2 y^2\=1\,\left[r\,\mathrm{\theta }\right]\right)\,0\,\mathrm{\π}\/4\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

12

$\mathrm{simplify}\left(\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(⟨0\,0⟩\,2\,1\,\left[r\,\mathrm{\theta }\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

13

$\mathrm{simplify}\left(\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Sector}\left(\mathrm{Ellipse}\left(⟨0\,0⟩\,2\,1\,\left[r\,\mathrm{\theta }\right]\right)\,0\,\mathrm{\π}\/4\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

14

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨0\,0⟩\,⟨1\,3⟩\,⟨4\,7⟩\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

Table 9.6.6   Examples of valid syntax for the Flux  command

Table 9.6.7   Comments on Table 9.6.6

Example 9.6.1

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on $S$, the surface of the rectangular parallelepiped (box) whose faces lie in the planes $x\=1\,x\=2\,y\=3\,y\=4\,z\=5\,z\=6$.

Example 9.6.2

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface of the sphere whose center is at the Cartesian point $\left(A\,B\,C\right)$ and whose radius is $a$.

Example 9.6.3

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface defined parametrically by the equations $x\left(u\,v\right)\=u\+v\,y\left(x\,v\right)\=u-v\,z\left(u\,v\right)\=u v$ and the bounds $u\in \left[0\,1\right]$, $v\in \left[0\,u\right]$.

Example 9.6.4

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface $z\=x^2\+y^2$ defined over the plane region $R$ bounded by $x\=0\,x\=\mathrm{\π}\/4\,y\=\sin \left(x\right)\,y\=\cos \left(x\right)$.

Example 9.6.5

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface $z\=x^2\+y^2$ defined over the unit disk with center at $\left(x\,y\right)\=\left(2\,3\right)$.

Example 9.6.6

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface $z\=x^2\+y^2$ defined over the rectangle whose edges lie in the lines $x\=1\,x\=2\,y\=1\,y\=3$.

Example 9.6.7

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface $z\=x^2\+y^2$ defined over the ellipse whose center is at $\left(x\,y\right)\=\left(2\,3\right)$, and whose semi-major and semi-minor axes are 1 and $1\/2$, respectively.

Example 9.6.8

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface $z\=x^2\+y^2$ defined over the triangle whose vertices $A\,B\,C$, are respectively $\left(1\,2\right)$, $\left(5\,1\right)$, and $\left(3\,3\right)$.

Example 9.6.9

Integrate the scalar $f\left(x\,y\,z\right)\=x y z$ on the surface $z\=x^2\+y^2$ defined over the first-quadrant part of the interior of the ellipse $x^2\+4 y^2\=1$ bounded by the lines $y\=0$ and $y\=x$.

Example 9.6.10

Obtain the flux of the field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ through the surface of the rectangular parallelepiped (box) whose faces lie in the planes $x\=1\,x\=2\,y\=3\,y\=4\,z\=5\,z\=6$.

Example 9.6.11

Working in spherical coordinates, obtain the flux of the field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ through the surface of the unit sphere centered at the origin. Use a parametric representation of the surface.

Example 9.6.12

Working in Cartesian coordinates, obtain the flux of the field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ through the surface of the unit sphere centered at the origin. Use a Cartesian representation of the surface.

Example 9.6.13

Obtain the flux of the field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ through the surface defined parametrically by the equations $x\left(u\,v\right)\=u\+v\,y\left(x\,v\right)\=u-v\,z\left(u\,v\right)\=u v$ and the bounds $u\in \left[0\,1\right]$, $v\in \left[0\,u\right]$.

Example 9.6.14

Obtain the flux of the field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ through the surface $z\=x^2\+y^2$ defined over the disk with center at $\left(x\,y\right)\=\left(1\,2\right)$ and with radius 3.

Example 9.6.15

Obtain the flux of the field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ through the surface $z\=x^2\+y^2$ defined over the interior of the ellipse $x^2\+4 y^2\=1$.

Example 9.6.16

Obtain the flux of the field $\mathbf{F}\=y z \mathbf{i}\+x z \mathbf{j}\+x y \mathbf{k}$ through the surface $z\=x^2\+y^2$ defined over the interior of the triangle whose vertices are $\left(1\,2\right)$, $\left(5\,1\right)$, and $\left(3\,3\right)$.