Understanding Stokes Theorem and is the variation of the metric a tensor?

In summary, the conversation between Haus and Greg discusses two questions regarding mathematics. The first question is about whether the variation of the metric is a tensor and how to express the covariant derivative of the variation using a test function. The second question is about understanding Stoke's theorem and its application for covariant divergences, partial derivatives, and exterior derivatives. Haus also mentions finding out more about these questions, which could provide further insight for those reading the thread.
  • #1
haushofer
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Hi, I have 2 little questions and hope to find some clarity here. It concerns some mathematics.

1) Is the variation of the metric again a tensor? I have the suspicion that it's not, because i would say that it doesn't transform like one. How can i get a sensible expression then for the covariant derivative of the variation of the metric? I need this to calculate the variation of the Riemanntensor, induced by a diffeomorphism multiplied by some test function.

2) I have the idea that I don't quite understand Stoke's theorem. Does it only apply for covariant divergences, or also for partial derivatives and/or exterior derivatives? I'm a little confused here :( I need this to rewrite the variation of an action.

Thanks in forward,

Haus.
 
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  • #3
Hi Greg,

Yes, I did, why do you ask? I didn't recognize the podt until I saw that it's 12 years old :D
 
  • #4
haushofer said:
Hi Greg,

Yes, I did, why do you ask? I didn't recognize the podt until I saw that it's 12 years old :D
Would you be interested in writing a couple sentences on each to give insight for those guests reading this thread? :smile:
 

1. What is Stokes Theorem and how is it used?

Stokes Theorem is a fundamental concept in vector calculus that relates the surface integral of a vector field to the line integral of its curl over the boundary of the surface. It is used to solve problems involving the flow of vector fields through surfaces.

2. What is the significance of using a tensor to understand Stokes Theorem?

A tensor is a mathematical object that represents the variation of a physical quantity with respect to changes in coordinate systems. In the context of Stokes Theorem, using a tensor allows us to account for the variation of the metric (i.e. the measurement of distances and angles) as we move along the boundary of the surface, which is necessary for accurately calculating the line integral.

3. How does the variation of the metric affect Stokes Theorem?

The variation of the metric is a crucial factor in understanding Stokes Theorem because it affects the calculation of the line integral over the boundary. As we move along the boundary, the metric may change, leading to different values for the line integral and ultimately affecting the overall result of the theorem.

4. Can you provide an example of using Stokes Theorem to solve a real-world problem?

One example of using Stokes Theorem to solve a real-world problem is in fluid dynamics, where it can be used to determine the circulation of a fluid flow around a surface. This information is essential in designing efficient aerodynamic structures, such as airplane wings.

5. How does Stokes Theorem relate to other fundamental theorems in vector calculus?

Stokes Theorem is closely related to other fundamental theorems in vector calculus, such as the Divergence Theorem and Green's Theorem. These theorems are all based on the concept of flux, or the flow of a vector field through a surface or along a curve. Stokes Theorem can be seen as a generalization of these other theorems, as it applies to both surfaces and curves.

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