Index Transformation: Understanding Notation & Index Rules

In summary: The index notation for differentiation is just a fancy way of saying \frac {D}{Dt} \omega = \omega_j \partial_j v_i \cdot \omega + \nu \partial_j \partial_j \omega_i \cdot \omega The first equation is just a shorthand for \frac{D \omega}{Dt} \omega_j = \omega_j \partial_j v_i + \nu \partial_j \partial_j \omega_i And the second equation is just a shorthand for \frac{D}{Dt
  • #1
Peregrine
22
0
In my struggles to understand index notation, I am trying to figure out how my book came up with the following transformation.

[tex] \frac {D \omega}{Dt} \cdot \omega = \omega_j \partial_j v_i \cdot \omega + \nu \partial_j \partial_j \omega_i \cdot \omega [/tex]
=
[tex] \frac {D \frac{\omega^2}{2}}{Dt} = \omega_i \omega_j S_{ji} + \nu \partial_j \partial_j \frac {\omega^2}{2} - \nu \partial_j \omega_i \partial_j \omega_i[/tex]

This is how far I got.

changing [tex] \frac {D \omega}{Dt} \cdot \omega[/tex] to index notation yields

[tex] \omega_i \partial_o \omega_i + \omega_i v_j \partial_j \omega_i = \omega_i \omega_j \partial_j v_i + \omega_i \nu \partial_j \partial_j \omega_i [/tex]

Looking back to the book's solution, this appears to say [tex]\omega_i \omega_i = 1/2 \omega^2[/tex]? I thought it would be simple [tex]\omega^2[/tex]?

Then, it is pretty clear to me that [tex]\omega_i \omega_j \partial_j v_i = S_{ji}[/tex], so the term

[tex] \omega_i \omega_j \partial_j v_i = \omega_i \omega_j S_{ji} [/tex]

Finally, looking at [tex] \omega_i \nu \partial_j \partial_j \omega_i [/tex], I get, from the chain rule,

[tex] \nu \omega_i \omega_i \partial_j \partial_j + \nu \omega_i \partial_j \partial_j \omega_i [/tex]

which can be rewritten as

[tex] \nu \partial_j \partial_j \omega_i \omega_i + \nu \partial_j \omega_i \partial_j \omega_i [/tex]

So again, I run into [tex]\omega_i \omega_i = 1/2 \omega^2[/tex], which I don't understand.

First, have I don this correctly, and second, any ideas? Thanks much.
 
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  • #2
Can you perhaps explain your notation? For example, in your first equation you are using capital Ds for differentiation (I assume that there is some particular reason for this). So, if you can answer the following questions then somebody here should be able to help you:

1) What's the significance of the upper case D in the differentiation?
2) What types of objects are [itex]\omega[/itex], [itex]v[/itex], and [itex]S[/itex]?
3) What's your understanding of the summation rule? For instance, in the first equation you have a term in which three quantities all have the same index. Given that it seems that both [itex]\omega[/itex] and [itex]v[/itex] are either one-forms or vectors, this term doesn't make sense.
 
  • #3
1. Capital D refers to the substantial derivative, in the notation of Stokes. It boils down to:

[tex] \frac {D()}{Dt} = \frac {\partial ()}{\partial t} + v_i \partial_i () [/tex]

Also, in the index notation of my book, [tex] \frac {\partial ()}{\partial t} = \partial_o [/tex]

2. w is a vector. v is a vector. And Sij is a 2nd order tensor.

3. You caught a typo in my writing. The term you speak of should have been [tex] \omega_j \partial_j v_i \cdot \omega [/tex]. I corrected this in the initial post.

Also, I do understand that the first equation I gave mixes index and symbolic notation, but that's how the book presented it, even though it seems to me to be bad form.

Thanks for taking a look at this. I understand this is probably a pretty simplistic problem for most here, but I am having a tough time understanding.
 
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  • #4
Okay, so looking at this backwards makes sense.

Since w is general, and thus can be f(x,y,z,t)

[tex] \frac {D \frac{\omega^2}{2}}{Dt}[/tex] differentiated by the chain rule gives:

[tex] \frac{2 \omega}{2} \frac{D \omega}{Dt}[/tex]

[tex] = \omega \cdot \frac{D \omega}{Dt}[/tex]

So, looking at it backwards makes it apparent that

[tex]\omega_i \frac{D}{Dt} \omega_i = \omega \cdot \frac {D \omega}{Dt}[/tex]

But I still don't see the transformation in index notation.
 

Related to Index Transformation: Understanding Notation & Index Rules

1. What is an index in mathematics?

An index in mathematics is a number or symbol used to indicate the position or order of an element in a sequence, matrix, or other mathematical structure. It is usually written as a subscript or superscript, and it helps to identify and distinguish individual elements within a larger set or formula.

2. How do you simplify indices?

Simplifying indices involves using index rules to rewrite an expression with indices into a simpler form. Some common index rules include the power rule (ab)^n = a^n * b^n, the quotient rule (a/b)^n = a^n / b^n, and the product rule (a^n)^m = a^(n*m). These rules can be used to combine or separate indices, and to convert between negative and fractional indices.

3. What is the difference between base and index?

The base is the main number or expression in an index, and the index is the number or symbol written as a superscript or subscript that indicates the power or root that the base is raised to. For example, in the expression 2^3, 2 is the base and 3 is the index. In the expression √16, 16 is the radicand (equivalent to the base) and 2 is the index.

4. How do you handle negative and fractional indices?

To handle negative indices, use the rule a^-n = 1/a^n, which states that a negative index is equivalent to the reciprocal of the base raised to the positive index. For fractional indices, use the rule a^(m/n) = (n√a)^m, which states that a fractional index is equivalent to taking the root of the base and then raising it to the positive power. It is also important to remember that the denominator of the fractional index represents the root, while the numerator represents the power.

5. What is the purpose of index transformation?

The purpose of index transformation is to simplify expressions and make them easier to work with. By using index rules to manipulate indices, we can rewrite complex expressions into simpler forms, which can help with solving equations, finding patterns, and making calculations more efficient. Index transformation is also useful for understanding and interpreting mathematical concepts and structures that involve indices, such as geometric sequences, matrices, and exponential functions.

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