What is the meaning of Green's Theorem via Curl?

In summary, the conversation revolved around difficulties in understanding Green's theorem, specifically the visualization of the Curl of a vector field. The individual was trying to view it from a physics perspective but struggled to make sense of a picture in their textbook showing a plane with region D in R^3, a vector k normal to D, and vectors F(x,y) pointing in a CCW direction around k. They requested a better visualization or explanation of the image.
  • #1
skepticboat
1
0
Hi Everyone,

Our class just learned Green's theorem using the Curl of a vector field F. I'm just having a tough time visualizing what in the world this means. I'm trying to view it in a physics perspective but I'm having a tough time. There's a picture in my textbook which has a plane with region D in R^3 with vector k which is normal to D. (Sorry I will definitely learn how to use this forum's programs to post pretty pictures of my symbolism). There are vectors F(x,y) on this plane D and they point to their respective directions. They seem to 'rotate' or point in a CCW direction around the normal K. Can anyone give me a better visualization (or decryption of this image) of Green's theorem? Thank-you!
 
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  • #2
I hope this will help:

 

Related to What is the meaning of Green's Theorem via Curl?

1. What is Green's Theorem via Curl?

Green's Theorem via Curl, also known as the Kelvin-Stokes theorem, is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a double integral over the region enclosed by the curve. It states that the line integral of a two-dimensional vector field around a closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve.

2. What is the purpose of Green's Theorem via Curl?

The purpose of Green's Theorem via Curl is to provide a powerful tool for calculating line integrals and double integrals in two-dimensional vector fields. It allows for the conversion of a difficult line integral into an easier double integral, making calculations more efficient and accurate.

3. How is Green's Theorem via Curl applied?

To apply Green's Theorem via Curl, the line integral is first converted into a double integral by taking the curl of the vector field. Then, the double integral is evaluated over the region enclosed by the curve. The resulting value is equal to the original line integral.

4. What are the conditions for Green's Theorem via Curl to hold?

In order for Green's Theorem via Curl to hold, the curve must be simple and closed, meaning it does not intersect itself and encloses a single region. Additionally, the vector field must be continuous and have continuous partial derivatives within the region enclosed by the curve.

5. What are some practical applications of Green's Theorem via Curl?

Green's Theorem via Curl is used in various fields such as physics, engineering, and economics to solve problems involving circulation and flux. It is also used in fluid mechanics to calculate the flow of fluids around closed curves. Additionally, it has applications in computer graphics for modeling and rendering of vector fields.

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