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- Calculus volume 3
- Vector calculus
- Green’s theorem
Key concepts
- Green’s theorem relates the integral over a connected region to an integral over the boundary of the region. Green’s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.
- Green’s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is
In the flux form, the integrand is
- Green’s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.
- A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.
Key equations
-
Green’s theorem, circulation form
where
C is the boundary of
D
-
Green’s theorem, flux form
where
C is the boundary of
D
-
Green’s theorem, extended version
For the following exercises, evaluate the line integrals by applying Green’s theorem.
where
C is the path from (0, 0) to (1, 1) along the graph of
and from (1, 1) to (0, 0) along the graph of
oriented in the counterclockwise direction
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where
C is the boundary of the region lying between the graphs of
and
oriented in the counterclockwise direction
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where
C is the boundary of the region lying between the graphs of
and
oriented in the counterclockwise direction
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where
C is the boundary of the region lying between the graphs of
and
oriented in the counterclockwise direction
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where
C consists of line segment
C
1 from
to (1, 0), followed by the semicircular arc
C
2 from (1, 0) back to (1, 0)
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For the following exercises, use Green’s theorem.
Evaluate line integral
where
C is the boundary of the region between circles
and
and is a positively oriented curve.
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Find the counterclockwise circulation of field
around and over the boundary of the region enclosed by curves
and
in the first quadrant and oriented in the counterclockwise direction.
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Calculate
where
C is a circle of radius 2 centered at the origin and oriented in the counterclockwise direction.
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Calculate integral
along triangle
C with vertices (0, 0), (1, 0) and (1, 1), oriented counterclockwise, using Green’s theorem.
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Questions & Answers
Explain the following terms .
(1) Abiotic factors in an ecosystem
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
Qasim
passive process of transport of low-molecular weight material according to its concentration gradient
AI-Robot
what is production?
Catherine
how did the oxygen help a human being
how did the nutrition help the plants
Biology is a branch of Natural science which deals/About living Organism.
evolutionary history and relationship of an organism or group of organisms
AI-Robot
cell is the smallest unit of the humanity biologically
Abraham
HOW CAN MAN ORGAN FUNCTION
the diagram of the digestive system
allimentary cannel
Ogenrwot
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is respiration
Deborah
Genetics is the study of heredity
Misack
how does twins formed?
Misack
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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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