Green’s theorem
Green’s theorem provides a connection between path integrals over a well-connected region in the plane and the area of the region bounded in the plane. Given a closed path bounding a region with area , and a vector-valued function over the plane,
where is the derivative of with respect to the th variable.
Corollary:
The closed path integral over a gradient of a function with continuous partial derivatives is always zero. Thus, gradients are conservative vector fields. The smooth function is called the potential of the vector field.
Proof:
The corollary states that
We can easily prove this using Green’s theorem.
But since this is a gradient…
Since for any function with continuous partials, the corollary is proven.
Title | Green’s theorem |
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Canonical name | GreensTheorem |
Date of creation | 2013-03-22 12:15:55 |
Last modified on | 2013-03-22 12:15:55 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 10 |
Author | mathcam (2727) |
Entry type | Theorem |
Classification | msc 26B20 |
Related topic | GaussGreenTheorem |
Related topic | ClassicalStokesTheorem |