Vector calculus identity using index and comma notation

In summary: I don't think I've ever seen that notation before. I don't think it's standard anywhere. It's just something that somebody came up with to make the index notation look a little nicer.I don't know - it just seems like a needless complication.
  • #1

hotvette

Homework Helper
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Homework Statement


Use index and comma notation to show:
\begin{equation*}\text{div }(\text{curl } \underline{\bf{v}}) = 0\end{equation*}

Homework Equations


\begin{align*}
& \text{(1) div } \underline{\bf{v}} = v_{i,i} \\
& \text{(2) curl } \underline{\bf{v}} = \epsilon_{ijk} v_{j,i} \underline{\bf{e}}_k
\end{align*}

The Attempt at a Solution


Substituting (2) into (1) I get:
\begin{equation*}
\text{(3) div }(\text{curl } \underline{\bf{v}}) = (\epsilon_{ijk} v_{j,i} \underline{\bf{e}}_k)_{p,p}
\end{equation*}
Don't know what to do next. I know the right hand side of (3) is correct because if I write out the individual terms the result is zero. But, we're not suppose to write out the terms. So, how does one tell that the right hand side of (3) is zero? Is there some special trick to interpret the expression?

P.S. I also have to show that [itex]\text{curl(grad }\phi)=\underline{\bf{0}}[/itex] but the question is really the same. How to interpret a (complex) index-comma expression without writing out the terms?
 
Last edited:
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  • #2
I think what's bogging you down is awful notation - I don't know why people use the comma notation but I find it much harder to spot and do index gymnastics with it as compared to more clear-cut differential operators.

hotvette said:
Substituting (2) into (1) I get:
\begin{equation*}
\text{(3) div }(\text{curl } \underline{\bf{v}}) = (\epsilon_{ijk} v_{j,i} \underline{\bf{e}}_k)_{p,p}
\end{equation*}
This can be simplified to ##(\epsilon_{ijp} v_{j,i} )_{,p}##, because the p-th component of the the curl is ##(\epsilon_{ijp} v_{j,i} )##. The next step then is to use the anti-symmetric property of the Levi-Civita tensor together with the symmetric nature of mixed partial derivatives to do some index relabelling to show that the term must be zero.
 
  • #3
Thanks, I can sort of see it. I guess it takes a lot of practice to begin to see how these things work! Sure wish there were some tutorials in internet land illustrating how to simplify index-comma expressions. Seems to me we need to see lots of examples in order to get the hang of things.
 
  • #4
Yes, the notation can be very powerful - but it takes time to figure out the tricks involved in performing the index gymnastics. Perhaps you could try picking up a book on tensor analysis (preferably geared towards physicists) or GR? Usually the first chapter or two will have plenty of examples and exercises on these.
 
  • #5
This subject is extremely confusing, and the terminology doesn't help. I think whoever coined [itex]\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j[/itex] as "tensor product" purposely wanted to confuse beginners of the subject. For days I thought "tensor product" was some mysterious operation on two vectors but could not find a definition of what the operation does. Even my professor couldn't easily explain what it meant (to someone unfamiliar with the subject). I slowly came to realize (still not sure) that it is simply a way to represent a collection of basis vectors, thus [itex]\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j[/itex] is a thing rather than an operation on two vectors.
 
Last edited:

What is the definition of a vector calculus identity using index and comma notation?

A vector calculus identity using index and comma notation is a mathematical expression that describes the relationship between vector quantities using index notation (e.g. vi) and comma notation (e.g. ∇, v). It is used to simplify and manipulate vector equations in a concise and efficient way.

What are some common vector calculus identities using index and comma notation?

Some common vector calculus identities using index and comma notation include the product rule (∇, (uv) = u∇v + v∇u), the chain rule (∇, f(g) = f'(g)∇g), and the gradient of a scalar field (∇, φ = ∂φ/∂xi).

Why is index and comma notation useful in vector calculus?

Index and comma notation allows for the concise representation and manipulation of vector equations, making it easier to perform vector operations and solve complex problems. It also allows for easier visualization and interpretation of vector quantities.

What are some applications of vector calculus identities using index and comma notation?

Vector calculus identities using index and comma notation are used in a variety of fields, including physics, engineering, and mathematics. They are particularly useful in solving problems involving motion, forces, and fields, such as in fluid dynamics, electromagnetism, and mechanics.

Can vector calculus identities using index and comma notation be applied to higher dimensions?

Yes, vector calculus identities using index and comma notation can be applied to any number of dimensions. The notation remains the same, but the equations may become more complex. This allows for the use of these identities in a wide range of mathematical and scientific contexts.

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