Understanding Greene's, Stoke's, and the Divergence theorems

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In summary: X is equal to the differential of the form over X)In summary, the Divergence Theorem, Green's Theorem, and Stokes' Theorem are all extensions of the fundamental theorem of calculus to multi-dimensional space. They allow for the evaluation of integrals over regions by considering the behavior at the boundaries of those regions. Each theorem deals with a different type of integral (divergence, flux, and curl) and applies to different dimensional regions (volume, surface, and curve). These theorems are important in understanding and solving problems in physics, particularly in E&M. It is common for students to struggle with these concepts, but with practice and a good understanding of differential forms, they can be mastered
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thatONEguy94
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Hi everyone, first post. Anyway, I am reviewing my math physics, and I am having trouble understanding the Divergence Theorem, Green's Theorem, and Stokes' Theorem. I was able to satisfactorily pass math physics by only being able to regurgitate them, but soon I will be taking e&m, and it seems that these theorems are very important to it.
Anyway, I get grad, div, and curl, but I am confused about these theorems so I will state what I think they are, and hopefully someone will correct me where I'm wrong.

First off Green's Theorem:
∮CLdx+Mdy=∫D∫(∂M∂x−∂L∂y)dydx
so to my understanding, the closed path integral about two functions defining a closed area C is equivalent to the double integral of that area.

Then with the Divergence Theorem:
∫∫∫V(∇⋅F⃗ )dV=∫S∫F⃗ ⋅dS⃗
To my understanding, the triple integral of the divergence of vector F through a three dimensional region is equivalent to the cross product of the vector F and the unit vector normal to the the surface of a the three dimensional region. It is my understanding that The Divergence theorem is Green's Theorem extended to three dimensions, and deals with a closed three dimensional region bounded by a surface that can be collapsed to two dimensions, while Green's theorem deals with a two-dimensional region outlined by two one-dimensional functions. Where I'm confused is that the divergence theorem seems to deal with a flow through the 3 dimensional area while Greens does not.

Finally for Stokes' Theorem:
∮CF⃗ ⋅dr⃗ =∫S∫∇×F⃗ ⋅dS⃗
my understanding is that it deals with a two dimensional surface in 3-dimensional space that is bounded by a 1-dimensional curve (my textbook used the example of a butterfly net). However, I am really confused by this theorem, and I don't even have an idea of what it means.

I know my ideas here are way off, but my textbook is really not of much help, so any clarification anyone could give me would be really helpful!
Thanks.
 
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  • #3
So after watching Khan Academy's Videos for the three of them (much more helpful than my miserable textbook), I think I understand it a lot better. Would I be correct in saying that Green's theorem allows one to evaluate a line integral in two dimensional space, the Divergence theorem allows you to evaluate a surface integral in three dimensional space, and stokes theorem allows you to evaluate a line integral in three dimensional space. Would I be correct in saying this??
 
  • #4
First, let me say that your "struggle" grasping these concepts is normall. Most students coming out of their first vector calculus course are in a similar position. Taking E&M and other vector based physics courses will help. And a good E&M prof will spend some time reviewing these concepts when the arise in the course.

That being said, do you remember the fundamental theorem of calculus?

If [itex]F' \left( x \right) = f\left(x \right) [/itex]

Then

[itex]\int_a^b f\left(x \right) dx = F(b)-F(a) [/itex]

By now you probably use this theorem routinely without thinking about it, but basically this theorem allows you to evaluate an integral over a region simply by looking at the behavior at the boundary.

It turns out that Green's Theorem, the Divergence Theorem, and Stokes Theorem are extension extension of the fundamental theorem of calculus to multi-dimension space. And they do exactly the same thing.

The divergence theorem allows you the evaluate you the integral of the divergence over the volume of a region by considering integral of the flux over the surface (or boundary) of that volume.

Stokes Theorem allows you to calculate the integral of the curl over an area by performing a line integral along the boundary.
(Green's theorem is actually just a simple case of Stokes Theorem).
 
  • #5
learn differential forms, then it says the integral of w over ∂X equals the integral of dw over X,
 

Related to Understanding Greene's, Stoke's, and the Divergence theorems

What is the purpose of Greene's, Stoke's, and the Divergence theorems?

The purpose of these theorems is to provide a mathematical framework for understanding the relationship between the surface and volume integrals of a vector field in three-dimensional space. They are fundamental tools in the study of vector calculus and are used to solve problems in physics, engineering, and other fields.

What is the difference between Greene's, Stoke's, and the Divergence theorems?

Greene's theorem relates line integrals to double integrals, Stoke's theorem relates surface integrals to line integrals, and the Divergence theorem relates volume integrals to surface integrals. They all have different applications and are used to solve different types of problems.

How are these theorems derived?

Greene's theorem can be derived from the fundamental theorem of calculus, Stoke's theorem is a special case of the generalized Stokes' theorem, and the Divergence theorem can be derived from the Gauss-Ostrogradsky theorem. All of these theorems are based on the concept of the fundamental theorem of calculus and involve manipulating integrals to relate them to each other.

What are the practical applications of these theorems?

These theorems have numerous practical applications in physics, engineering, and other fields. For example, they can be used to calculate fluid flow, electric and magnetic fields, and heat transfer in various systems. They are also essential in the study of fluid and solid mechanics, electromagnetism, and thermodynamics.

What are the limitations of these theorems?

Greene's, Stoke's, and the Divergence theorems have specific conditions and assumptions that must be met in order to apply them. For example, Greene's theorem applies only to simple closed curves, Stoke's theorem applies only to smooth surfaces, and the Divergence theorem applies only to regions with continuous boundaries. In addition, these theorems are limited to three-dimensional space and may not be applicable in higher dimensions.

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