The Idea Behind Green’s Theorem
Math Insight
When C is an oriented1 simple2 closed3 curve, the integral
represents the circulation4 of F around C. If F were the velocity field of water flow,
for example, this integral would indicate how much the water tends to circulate
around the path in the direction of its orientation.
One way to compute this circulation is, of course, to compute the line
integral5 directly. But, if our line integral happens to be in two dimensions
(i.e., F is a two-dimensional vector field and C is a closed path that lives in the
plane), then Green's theorem applies and we can use Green's theorem as an
alternative way to calculate the line integral.
Green's theorem transforms the line integral around C into a double integral6
over the region inside C. However, it's not obvious what function we should
integrate over the region inside C so that we still get the same answer as the line
integral. The notion of circulation can aid us in determining what this function
should be.
Think of the integral ∫CF⋅ds as the “macroscopic” circulation of the vector
field F around the path C. Now, imagine you came up with a “microscopic”
version of circulation around a curve. This microscopic circulation at a
point (x,y) has to tell you how much F would circulate around a tiny closed curve
centered around (x,y). We could picture the microscopic circulation as a bunch of
small closed curves (shown below in green), where each curve represents the
tendency for the vector field to circulate at that location (imagine that the small
curves were really, really small, much smaller than pictured).
Green's theorem is simply a relationship between the macroscopic circulation
around the curve C and the sum of all the microscopic circulation that is inside C.
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Attributed to: [Duane Q. Nykamp]
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If C is a simple closed curve in the plane (remember, we are talking about two
dimensions), then it surrounds some region D (shown in red) in the plane. D is
the “interior” of the curve C.
Green's theorem says that if you add up all the microscopic circulation
inside C (i.e., the microscopic circulation in D), then that total is exactly the same
as the macroscopic circulation around C.
“Adding up” the microscopic circulation in D means taking the double integral of
the microscopic circulation over D. Therefore, we can write Green's theorem as
What is this microscopic circulation? The microscopic circulation of Green's
theorem is the same as the microscopic circulation of the curl7 of a threedimensional vector field. The only difference is that Green's theorem applies only
with two-dimensional vector fields, e.g., for vector fields in the xy-plane. The
microscropic circulation we want is circulation in the xy-plane.
Microscopic circulation in the xy plane turns out to be the z-component of the
curl. You can see this as follows. The direction of the curl7 and the definition of its
components8 is determined by the right-hand rule. (Imagine curling the fingers of
your right hand around the circles indicating the circulation. One represents such
circulation by a vector pointing in the direction of your thumb.) In threedimensions, point the thumb of your right hand in the positive z direction. Then,
the fingers of your right hand curl in the counterclockwise direction parallel to
the xy-plane. So, the right-hand rule says that circulation in the xy-plane should
correspond to the z-component of the curl.
We conclude that, for Green's theorem,
“microscopic circulation”=(curlF)⋅k,
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(where k is the unit vector9 in the z-direction) and we can write Green's theorem
as
The component of the curl8 in the z-direction is given by the formula
This z-component of the curl is often termed the scalar curl of a two-dimensional
vector field. You can get intuition8 behind this formula or see how to derive10 this
formula from the definition of the curl11. Using this formula, we can write Green's
theorem as
To make sure that Green's theorem gives the right answer, we need to be careful
how we orient the curve C. The right hand rule says that (curlF)⋅k corresponds to
the amount of circulation in the counterclockwise direction. Hence, Green's
theorem, as we have written it, is valid only for curves oriented counterclockwise
(as pictured above). In this case, we say that C is a positively oriented boundary
of the region D. One way to remember what positively oriented means is the
following: if you were to walk alongC in the positive orientation, the region D will
be to your left. If you mess up the orientation, you'll be off by a minus sign.
Green's theorem and other fundamental theorems
Green's theorem is one of the four fundamental theorems of vector calculus12 all
of which are closely linked. Once you learn about surface integrals13, you can
see how Stokes' theorem14 is based on the same principle of linking microscopic
and macroscopic circulation.
What if a vector field had no microscopic circulation? Looking at Green's
theorem, we immediately see if this microscopic circulation
in some region D, then the line integral
were zero
for any closed curve C in D (for example, the closed curve C that is the boundary
of D). This is a property15 of conservative or path-independent16 vector fields,
which forms the basis for the gradient theorem17 for line integrals.
Are you ready to use Green's theorem? Make sure you understand when you
are allowed to use Green's theorem18, check out some other ways of writing
Green's theorem19, and then investigate some examples20. To become a master
Source URL: http://mathinsight.org/greens_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
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at Green's theorem, you should understand how it applies to more
general regions with holes21.
Source URL: http://mathinsight.org/greens_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
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Notes and Links:
1. http://mathinsight.org/oriented_curve_definition
2. http://mathinsight.org/simple_curve_definition
3. http://mathinsight.org/closed_curve_definition
4. http://mathinsight.org/line_integral_circulation
5. http://mathinsight.org/line_integral_vector_field_introduction
6. http://mathinsight.org/double_integral_introduction
7. http://mathinsight.org/curl_idea
8. http://mathinsight.org/curl_components
9. http://mathinsight.org/unit_vector_definition
10. http://mathinsight.org/circulation_unit_area_calculation
11. http://mathinsight.org/curl_definition_line_integral
12. http://mathinsight.org/fundamental_theorems_vector_calculus_summary
13. http://mathinsight.org/surface_integral_vector_field_introduction
14. http://mathinsight.org/stokes_theorem_idea
15. http://mathinsight.org/conservative_vector_field_no_circulation
16. http://mathinsight.org/conservative_vector_field_introduction
17. http://mathinsight.org/gradient_theorem_line_integrals
18. http://mathinsight.org/greens_theorem_when_applies
19. http://mathinsight.org/greens_theorem_other_ways_writing
20. http://mathinsight.org/greens_theorem_examples
21. http://mathinsight.org/greens_theorem_multiple_boundary_components
Source URL: http://mathinsight.org/greens_theorem_idea
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 5 of 5