MA 242.003

Day 68: April 22, 2013•Green’s theorem example•13.7: Stokes’ theorem examples•13.8: The Divergence Theorem

13.4: Green’s TheoremLet C be a positively oriented, piecewise-smooth, simple closed curved in the plane, and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region containing D, then

Application of Green’s theorem: Area of a plane region

QUESTION: Since the AREA of a plane region D is given by the double integral of f(x,y) = 1 over D, can we choose P(x,y) and/or Q(x,y) in Greens’ theorem to give the area of D?

Application of Green’s theorem: Area of a plane region

QUESTION: Since the AREA of a plane region D is given by the double integral of f(x,y) = 1 over D, can we choose P(x,y) and/or Q(x,y) in Greens’ theorem to give the area of D?

ANSWER: Yes, choose Q = x and P=0, so then

Application of Green’s theorem: Area of a plane region

QUESTION: Since the AREA of a plane region D is given by the double integral of f(x,y) = 1 over D, can we choose P(x,y) and/or Q(x,y) in Greens’ theorem to give the area of D?

ANSWER: Yes, choose Q = x and P=0, so then

Example: Compute the area of a circle of radius a

Parameterization of the circle:

Example: Compute the area of a circle of radius a

Parameterization of the circle:

Area =

(continuation of example)

(continuation of example)

Section 13.7

The Divergence Theorem of Gauss

Section 13.7

The Divergence Theorem of Gauss

Relates a flux integral of a vector field to the volume integral of the divergence of that vector field.

First we need a definition:

First we need a definition:

Definition: A solid 3-dimsional region is SIMPLE if it can be described as a type 1, type 2 and a type 3 region in space.

First we need a definition:

Definition: A solid 3-dimsional region is SIMPLE if it can be described as a type 1, type 2 and a type 3 region In space.

Note: E and S are uniquely related to each other, unlike the relationshipbetween S and C in Stokes’ theorem.

The divergence theorem is used, for example, in electrostatics, where one encloses a region inside a “Gaussian pillbox” as in the example below:

Here is a clip from Wikipedia which discusses “valid” and “invalid” Gaussian surfaces.

Here is a clip from Wikipedia which discusses “valid” and “invalid” Gaussian surfaces.

A valid surface must enclose a 3-dimensional region E.

Remark about problem STATEMENTS:

Remark about problem STATEMENTS:

1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

Remark about problem STATEMENTS:

1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

then you should compute

Remark about problem STATEMENTS:

1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

then you should compute

2. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

Remark about problem STATEMENTS:

1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

then you should compute

2. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

then you should compute

(continuation of example)

(continuation of example)