Use the divergence theorem to calculate the flux of a vector field.
Apply the divergence theorem to an electrostatic field.
We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study. The divergence theorem has many uses in physics; in particular, the divergence theorem is used in the field of partial differential equations to derive equations modeling heat flow and conservation of mass. We use the theorem to calculate flux integrals and apply it to electrostatic fields.
Overview of theorems
Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the
Fundamental Theorem of Calculus we have discussed:
The Fundamental Theorem of Calculus :
This theorem relates the integral of derivative
over line segment
along the
x -axis to a difference of
evaluated on the boundary.
The Fundamental Theorem for Line Integrals :
where
is the initial point of
C and
is the terminal point of
C . The
Fundamental Theorem for Line Integrals allows path
C to be a path in a plane or in space, not just a line segment on the
x -axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative
over path
C to a difference of
evaluated on the boundary of
C .
Green’s theorem, circulation form :
Since
and curl is a derivative of sorts,
Green’s theorem relates the integral of derivative curl
F over planar region
D to an integral of
F over the boundary of
D .
Green’s theorem, flux form :
Since
and divergence is a derivative of sorts, the flux form of Green’s theorem relates the integral of derivative div
F over planar region
D to an integral of
F over the boundary of
D .
Stokes’ theorem :
If we think of the curl as a derivative of sorts, then
Stokes’ theorem relates the integral of derivative curl
F over surface
S (not necessarily planar) to an integral of
F over the boundary of
S .
Stating the divergence theorem
The divergence theorem follows the general pattern of these other theorems. If we think of divergence as a derivative of sorts, then the
divergence theorem relates a triple integral of derivative div
F over a solid to a flux integral of
F over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field
F over a closed surface
S to a triple integral of the divergence of
F over the solid enclosed by
S .
The divergence theorem
Let
S be a piecewise, smooth closed surface that encloses solid
E in space. Assume that
S is oriented outward, and let
F be a vector field with continuous partial derivatives on an open region containing
E (
[link] ). Then
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?