Chapter 9: Vector Calculus
Section 9.6: Surface Integrals
|
Essentials
|
|
The element of arc length for a curve defined by is .
Similarly, the element of surface area for a surface is , where Table 9.6.1 lists the forms of for surfaces defined in various ways. Derivations of these expressions are given in the Examples.
|
|
Explicit
|
|
|
Implicit
|
|
|
Parametric
|
|
, where
= =
= =
= =
|
Table 9.6.1 Forms of the surface-area element
|
|
|
For the explicit and implicit cases, or , depending on the order of integration chosen. For the parametric case, or , again, depending on the order of integration chosen.
The surface area of a surface is given by the double integral .
The surface integral of the scalar function taken over the surface is given by one of the double integrals listed in Table 9.6.2.
|
|
Explicit
|
|
|
Implicit
|
|
Parametric
|
|
|
Table 9.6.2 Surface integral of the scalar function
|
|
|
The surface integral of when is given implicitly can be implemented analytically provided the equation can be solved explicitly for .
The surface integral (over ) of the normal component of a vector field F is called the flux of F through ., and is given by the double integral .
|
|
Surface Integrals in Maple
|
|
The Student VectorCalculus package contains the SurfaceInt and Flux commands for integrating over surfaces. The SurfaceInt command integrates a scalar function over various pre-defined surfaces; the Flux command, the normal component of a vector field.
Table 9.6.3 lists task templates for surface integrals of scalars and vector fields.
The SurfaceInt command accepts as options the pre-defined regions Sphere and Box. Other surfaces can be defined parametrically by the Surface option, which itself allows the surface to be defined over any of the two-dimensional regions known to int, namely, Circle, Ellipse, Rectangle, Region, Sector, and Triangle. Table 9.6.4 contains a number of examples of the appropriate syntax for the SurfaceInt command applied to the scalar function . The optional "output" is set to integral; the default is value. (The value of the integral is also returned if is omitted.)
•
|
Install the Student VectorCalculus package
|
|
|
|
|
|
1
|
|
2
|
|
3
|
|
4
|
|
5
|
|
6
|
|
7
|
|
8
|
|
9
|
|
10
|
|
11
|
|
12
|
|
Table 9.6.4 Examples of valid syntax for the SurfaceInt command
|
|
|
If the Student VectorCalculus package is loaded, then the commands in Table 9.6.4 can be executed right in the table. Comments on the items in Table 9.6.4 are listed in Table 9.6.5.
•
|
Items (1) and (2) in Table 9.6.4 illustrate the basic syntax for integration over a sphere and the surface of a rectangular parallelepiped (box), respectively.
|
•
|
Items (3-12) show the use of the Surface option whereby a surface can be defined parametrically. (In Table 9.6.4, the parametrization is .)
|
•
|
Items (3) and (4) result in the same integral, with (4) showing how the Region option is implemented.
|
•
|
Items (5) and (6) again result in the same integral, with (6) showing how the Rectangle option is implemented.
|
•
|
Item (7) shows how the Circle option is implemented, and includes the names of the polar coordinates that this option invokes. (Otherwise, and would be used in place of and , respectively.)
|
•
|
Items (8) and (11) show how the Sector option can be applied to a circle or ellipse.
|
•
|
The Ellipse option also appears in items (9) and (10), the former defining the ellipse via an expression; the latter, via center and semi-major and semi-minor axes. (Note also the inclusion of the names of the polar variables that will be displayed in the unevaluated integral.)
|
•
|
The simplify command applied in items (9-11) makes the resulting returns much more readable.
|
•
|
Item (12) shows the syntax for defining a surface over a triangle determined by its three vertices.
|
|
Table 9.6.5 Comments on Table 9.6.4
|
|
|
Similarly, the Flux command accepts exactly the same options as the SurfaceInt command. In the Student VectorCalculus package, the Flux command with the options Sphere, Box, and Surface (without the pre-defined regions known to int) will draw a graph of the vector field, the surface, and a normal to the surface. For the closed surfaces defined by the Sphere and Box options, the additional parameters inward and outward are recognized, with outward being the default for the direction of the normal field on the surface. Table 9.6.6 contains a number of examples of the appropriate syntax for the Flux command applied to the vector field F. As in Table 9.6.4, the optional "output" is set to integral; the default is value. Numbered cells shaded in yellow indicate forms of the command that support the optional , provided the domain is a rectangle with edges parallel to the coordinate axes.
•
|
Install the Student VectorCalculus package
|
|
|
|
|
|
|
|
1
|
|
2
|
|
3
|
|
4
|
|
5
|
|
6
|
|
7
|
|
8
|
|
9
|
|
10
|
|
11
|
|
12
|
|
13
|
|
14
|
|
Table 9.6.6 Examples of valid syntax for the Flux
command
|
|
|
If the Student VectorCalculus package is loaded, then the commands in Table 9.6.6 can be executed right in the table. Comments on the items in Table 9.6.6 are listed in Table 9.6.7.
•
|
In items (1-3), the option will generate a graph of the field, the surface, and a normal vector.
|
•
|
Items (4) and (7) return the same integral. The domain need not be rectangular.
|
•
|
Items (5) and (6) return the same integral.
|
•
|
Item (8) defines a surface over a disk.
|
•
|
Item (10) defines a surface over the interior of an ellipse defined by an equation.
|
•
|
Item (12) defines a surface over the interior of an ellipse defined by its center and semi-major and semi-minor axes.
|
•
|
Items (9), (11), and (13) use the Sector option to define a surface over portions of a disk or ellipse.
|
•
|
Item (14) defines a surface over the interior of a triangle specified by its vertices.
|
|
Table 9.6.7 Comments on Table 9.6.6
|
|
|
|
|
Examples
|
|
Example 9.6.1
|
Integrate the scalar on , the surface of the rectangular parallelepiped (box) whose faces lie in the planes .
|
Example 9.6.2
|
Integrate the scalar on the surface of the sphere whose center is at the Cartesian point and whose radius is .
|
Example 9.6.3
|
Integrate the scalar on the surface defined parametrically by the equations and the bounds , .
|
Example 9.6.4
|
Integrate the scalar on the surface defined over the plane region bounded by .
|
Example 9.6.5
|
Integrate the scalar on the surface defined over the unit disk with center at .
|
Example 9.6.6
|
Integrate the scalar on the surface defined over the rectangle whose edges lie in the lines .
|
Example 9.6.7
|
Integrate the scalar on the surface defined over the ellipse whose center is at , and whose semi-major and semi-minor axes are 1 and , respectively.
|
Example 9.6.8
|
Integrate the scalar on the surface defined over the triangle whose vertices , are respectively , , and .
|
Example 9.6.9
|
Integrate the scalar on the surface defined over the first-quadrant part of the interior of the ellipse bounded by the lines and .
|
Example 9.6.10
|
Obtain the flux of the field through the surface of the rectangular parallelepiped (box) whose faces lie in the planes .
|
Example 9.6.11
|
Working in spherical coordinates, obtain the flux of the field 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 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 through the surface defined parametrically by the equations and the bounds , .
|
Example 9.6.14
|
Obtain the flux of the field through the surface defined over the disk with center at and with radius 3.
|
Example 9.6.15
|
Obtain the flux of the field through the surface defined over the interior of the ellipse .
|
Example 9.6.16
|
Obtain the flux of the field through the surface defined over the interior of the triangle whose vertices are , , and .
|
|
|
|
<< Previous Section Table of Contents Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|