Tensor calculus, dummy indices

In summary, the conversation discusses the possibility of renaming dummy indices in a tensor equation with separate terms. The equation in question involves \epsilon_{kpq} and \frac{\partial u_p}{\partial x_q}, and it is shown that the dummy indices can be changed in the separate terms to obtain the desired result of \frac{1}{2}\epsilon_{kqp}\frac{\partial u_p}{\partial x_q}. It is confirmed that this step is correct by taking separate sums for the different terms.
  • #1
Telemachus
835
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Hi there. When I have dummy indices in a tensor equation with separate terms, I wanted to know if I can rename the dummies in the separate terms.

I have, in particular:

[tex]\displaystyle w_k=-\frac{1}{4}\epsilon_{kpq}\left [ \frac{\partial u_p}{\partial x_q}-\frac{\partial u_q}{\partial x_p} \right ]=-\frac{1}{4}\epsilon_{kpq}\frac{\partial u_p}{\partial x_q}+\frac{1}{4}\epsilon_{kpq}\frac{\partial u_q}{\partial x_p}=\frac{1}{4}\epsilon_{kqp}\frac{\partial u_p}{\partial x_q}+\frac{1}{4}\epsilon_{kpq}\frac{\partial u_q}{\partial x_p} [/tex]

I've used that [tex]\epsilon_{kpq}=-\epsilon_{kqp}[/tex]

So, if I can change the dummy indices for the separate terms I can use that:

[tex]\frac{1}{4}\epsilon_{kpq}\frac{\partial u_q}{\partial x_p}=\frac{1}{4}\epsilon_{kqp}\frac{\partial u_p}{\partial x_q}[/tex]
to get:
[tex] w_k= \frac{1}{2} \epsilon_{kqp} \frac{\partial u_p}{\partial x_q} [/tex]

which is the result I'm looking for, but I wasn't sure if the last step is right.
 
Last edited:
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  • #2
mhm, I think that it suffices to just take separate sums for the different terms to see that yes, I just can change the dummies.
 

1. What is tensor calculus?

Tensor calculus is a branch of mathematics that deals with the study of tensors, which are mathematical objects that describe the relationships between different coordinate systems. It involves the use of mathematical operations and rules to manipulate tensors and solve problems in physics, engineering, and other fields.

2. What are dummy indices in tensor calculus?

Dummy indices are placeholders for indices that are summed over in a tensor expression. They are typically represented by Greek letters and are used to simplify and generalize tensor equations without specifying the specific values of the indices.

3. How are dummy indices used in tensor calculus?

Dummy indices are used to simplify and generalize tensor expressions by indicating which indices are summed over. They allow for more compact and efficient notation, as well as a more systematic way of solving problems involving tensors.

4. What is the difference between upper and lower dummy indices?

Upper and lower dummy indices represent different types of tensors in tensor calculus. Upper indices represent contravariant tensors, which describe how a vector changes with respect to changes in the coordinate system, while lower indices represent covariant tensors, which describe how a scalar quantity changes with respect to changes in the coordinate system.

5. How do I know when to use dummy indices in a tensor expression?

Dummy indices are typically used when there are repeated indices in a tensor expression, as this indicates a summation over those indices. In general, they are used to simplify and generalize tensor equations, but their usage may vary depending on the specific problem or context.

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