Green,s Theorem
المؤلف:
Kaplan, W
المصدر:
"Green,s Theorem." §5.5 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley
الجزء والصفحة:
...
29-9-2018
2780
Green's Theorem
Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a region 
in the plane with boundary 
, Green's theorem states
 |
(1)
|
where the left side is a line integral and the right side is a surface integral. This can also be written compactly in vector form as
 |
(2)
|
If the region 
is on the left when traveling around 
, then area of 
can be computed using the elegant formula
 |
(3)
|
giving a surprising connection between the area of a region and the line integral around its boundary. For a plane curve specified parametrically as 
for ![t in [t_0,t_1]](http://mathworld.wolfram.com/images/equations/GreensTheorem/Inline7.gif)
, equation (3) becomes
 |
(4)
|
which gives the signed area enclosed by the curve.
The symmetric for above corresponds to Green's theorem with 
and 
, leading to
However, we are also free to choose other values of 
and 
, including 
and 
, giving the "simpler" form
 |
(10)
|
and 
and 
, giving
 |
(11)
|
A similar procedure can be applied to compute the moment about the 
-axis using 
and 
as
 |
(12)
|
and about the 
-axis using 
and 
as
 |
(13)
|
where the geometric centroid 
is given by 
and 
.
Finally, the area moments of inertia can be computed using 
and 
as
 |
(14)
|
using 
and 
as
 |
(15)
|
and using 
and 
as
 |
(16)
|
REFERENCES:
Arfken, G. "Gauss's Theorem." §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.
Kaplan, W. "Green's Theorem." §5.5 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 286-291, 1991.
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