Chain rule A(r,t)

  • Thread starter Confundo
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  • #1
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Problem

Use the chain rule to proof
[tex]
\dot{A}=\partial_t A+v_j\partial_jA_i
[/tex]

Attempt at Solution

[tex]
\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i}

[/tex]

Obviously
[tex]
v_j = \frac{dr_j}{dt}
[/tex]

I'm puzzled where the v_j and partial d_j come in
 

Answers and Replies

  • #2
fzero
Science Advisor
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If you're talking about why the index is j and not i, the answer is simple. The problem is using the summation convention that a repeated index is understood to be summed over:

[tex]
v_j\partial_j \equiv \sum_j v_j\partial_j.
[/tex]

When you wrote

[tex]
\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i},
[/tex]

the index in the derivatives should not have been the same index on [tex]A_i[/tex], since it is confusing to have the same index appearing three times when it is being implicitly summed over in part of the expression. You should have written

[tex]

\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+ \sum_j \frac{dr_j}{dt}\frac{\partial A_i}{\partial r_j} = \partial_t A_i+ \frac{dr_j}{dt}\frac{\partial A_i}{\partial r_j},


[/tex]

where the summation convention is being used in the last expression.
 

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