Another vector problem

[neelakash]

Homework Statement

Consider two elementary parallelopiped in tandem (i.e. joined side by side).Calculate the flux of a vector field F through these two parallelopiped and arrive at Gauss's divergence theorem

Homework Equations

The Attempt at a Solution

All standard textbooks contain the proof with a single parallelopiped.I do not understand in what way this problem is going to be different.Please help if some new trick is to be employed

 

[anathema]

.

Thank you for your interesting question about calculating the flux of a vector field through two parallelopipeds and arriving at Gauss's divergence theorem. I understand that you are familiar with the proof for a single parallelepiped, but are unsure how to approach the problem with two in tandem.

Firstly, let's define some notation for clarity. Let the two parallelepiped be denoted as A and B, with their respective dimensions and orientations. The vector field F can be written as F(x,y,z) = (F1(x,y,z), F2(x,y,z), F3(x,y,z)).

To calculate the flux through A and B, we can use the same method as for a single parallelepiped. We divide each parallelepiped into small cubes and calculate the flux through each of these cubes. Then, we add up the flux through all the cubes to get the total flux through A and B.

However, in order to arrive at Gauss's divergence theorem, we need to consider the boundaries of the two parallelepiped. Since A and B are joined side by side, their common boundary will have a normal vector perpendicular to it. Let's call this normal vector n.

Using the divergence theorem, we know that the flux through the boundary of A and B is equal to the volume integral of the divergence of F over the enclosed volume. In other words, the flux through the boundary is equal to the net flow of the vector field out of the enclosed volume.

Therefore, by adding up the flux through all the cubes in A and B, we are essentially calculating the net flow of the vector field out of the enclosed volume, which is equal to the flux through the boundary.

I hope this helps to clarify the problem for you. Please let me know if you have any further questions or if you would like me to elaborate on any specific steps in the solution.

 

[m2003]

Thank you for your question. It seems like you are trying to calculate the flux of a vector field through two parallelepipeds joined side by side and use Gauss's divergence theorem to do so. This is a valid approach and can be done by considering the two parallelepipeds as a single larger parallelepiped and applying the divergence theorem to it.

However, if you are looking for a different or more efficient method, one approach could be to use the concept of superposition, where you can break down the vector field into two simpler fields and then calculate the flux through each individual parallelepiped. Then, by adding the two flux values together, you can arrive at the total flux through the two parallelepipeds. This method may be more efficient in certain cases and can also be extended to more than two parallelepipeds.

Additionally, you could also consider using the concept of boundary surfaces and applying the divergence theorem to each individual surface of the parallelepipeds. This may provide a more visual understanding of the problem and can also be extended to more complex shapes.

Overall, there are multiple approaches to solving this problem and it ultimately depends on the specific situation and the level of complexity desired. I hope this helps and good luck with your calculations!