Gauss Divergence Theorem - Silly doubt - Almost solved

In summary, the person asking the question is wondering if they can separate the x and y components of a vector and write them as separate equations. Another person responds that they can, but it is not trivial to prove it. They suggest using x and y as parameters and looking at the proof in a calculus textbook for more information.
  • #1
come2ershad
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Homework Statement



The problem statement has been attached with this post.

Homework Equations



I considered u = ux i + uy j and unit normal n = nx i + ny j.


The Attempt at a Solution



I used gauss' divergence theorem. Then it came as integral [(dux/dx) d(omega)] + integral [(duy/dy) d(omega)] = integral [(ux nx d(gamma)] + integral [(uy ny d(gamma)]

My question is can I separate the x and y components and write as separate equations as given in the problem? Is that right?
 

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  • #2
anybody?
 
  • #3
You can, although it is not trivial to prove it. Break up the boundary so that the components of the vector can be written as functions of x on each piece and use x itself as parameter. That will give the first equation.

Then break up the boundary so the components of the vector can be written as functions of y on each piece and use y itself as parameter. That will give the second equation.

That partitioning is used in the proof of the theorem. You might want to look at the proof in any Calculus text.
 
  • #4
Thanks for the reply.
 

FAQ: Gauss Divergence Theorem - Silly doubt - Almost solved

What is the Gauss Divergence Theorem and why is it important?

The Gauss Divergence Theorem, also known as Gauss's Theorem or the Divergence Theorem, is a fundamental concept in vector calculus. It states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field over the enclosed volume. This theorem is important because it allows us to easily calculate the flux of a vector field without having to evaluate the surface integral directly.

What are some real-world applications of the Gauss Divergence Theorem?

The Gauss Divergence Theorem has many practical applications in physics and engineering. One example is in fluid dynamics, where it is used to calculate the rate of flow of a fluid through a given surface. It is also used in electromagnetics to calculate the electric flux through a closed surface and in thermodynamics to determine the heat flow through a surface.

How does the Gauss Divergence Theorem relate to other theorems in calculus?

The Gauss Divergence Theorem is closely related to the Fundamental Theorem of Calculus and Green's Theorem. It can also be seen as a higher-dimensional version of the Divergence Theorem in two dimensions. These theorems all involve the relationship between integrals and derivatives, and understanding the connections between them can help to deepen our understanding of calculus.

Is there a simple way to visualize the Gauss Divergence Theorem?

One way to visualize the Gauss Divergence Theorem is to imagine a closed surface in three-dimensional space, with a vector field passing through it. The flux of the vector field through the surface can be thought of as the "net flow" of the field through the surface. The divergence of the vector field at any point can be thought of as the "source" or "sink" of the field at that point. The theorem states that the net flow through the surface is equal to the sum of all the sources and sinks within the enclosed volume.

Are there any limitations or assumptions associated with the Gauss Divergence Theorem?

The Gauss Divergence Theorem has some limitations and assumptions that must be considered when using it. It assumes that the vector field is continuous and differentiable within the enclosed volume, and that the surface is smooth and well-behaved. Additionally, the theorem only applies to closed surfaces, meaning that it cannot be used for open surfaces or surfaces with holes. It is also important to note that the theorem only holds in three-dimensional space.

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