Index Notation Help: Solving ∫∂k(gixiεjklxl)dV

In summary, the conversation is about integrating the expression ∂k(gixiεjklxl) dV and how to proceed with the derivative, with the use of the product rule instead of the chain rule. The larger problem involves a gravity vector g, treated as a constant and pulled outside the integral. The final solution is expected to have a form of a cross product with the need to figure out the content inside the parentheses. There is also a mention of potential incorrectness in the tensor notation used.
  • #1
squire636
39
0

Homework Statement



∫ ∂k(gixiεjklxl dV

Can anyone make sense of this? I know I'll need to apply the chain rule when taking the derivative, but I'm not quite sure how to proceed. Also, this is part of a larger problem where g is a gravity vector existing purely in the -z direction, but I treated this as a constant and pulled it outside the integral and got:

g ∫ ∂k(xiεjklxl) dV


Homework Equations





The Attempt at a Solution



I don't think we need to get rid of the integral at all, and I'm pretty sure that the final solution will have something of the form: ∫ x x ( ) dV (that is a cross product). The TA for the course said that we should get to this, and we need to figure out what is inside the parentheses.

Thanks!
 
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  • #2
There's no chain rule here; just the product rule. What's [itex]\partial_k x_i[/itex]?
 
  • #3
If this is tensor notation, then it is simply wrong, because repeated indices there occur all at the bottom. You will have to use LaTeX to make your problem understandable. Speaking of which, knowing what the problem is about would be useful.
 

1. What is the purpose of index notation in this equation?

Index notation is used in this equation to simplify and condense the expression. It allows for easier manipulation and understanding of equations involving vectors and tensors.

2. How do I interpret the symbols used in this equation?

The symbol ∫ represents the integral, ∂k represents the partial derivative with respect to the kth variable, gi represents the ith component of the vector g, xi represents the ith component of the vector x, εjkl represents the Levi-Civita symbol, and xl represents the lth component of the vector x.

3. What is the significance of the Levi-Civita symbol in this equation?

The Levi-Civita symbol is used to represent the orientation of a coordinate system. It is equal to 1 if the indices are in ascending order, -1 if they are in descending order, and 0 if any indices are repeated. In this equation, it is used to simplify the expression and account for different orientations of the coordinate system.

4. How do I solve this equation?

To solve this equation, first take the partial derivative with respect to the kth variable, then use the Levi-Civita symbol to simplify the expression. Next, integrate over the volume, using the limits of integration and any necessary change of variables. Finally, evaluate the resulting expression to obtain your solution.

5. What are some real-world applications of index notation?

Index notation is commonly used in fields such as physics and engineering, where equations involving vectors and tensors are frequently encountered. It is also used in computer science, particularly in fields related to graphics and image processing. Additionally, it is used in mathematical fields such as differential geometry and tensor calculus.

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