Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I recently started a course on continuum mechanics. It started with the mathematical background of transforming tensors with contravariant and/or covariant indices. There is one thing I don't understand and it should be really straight forward. I hope you can give me a hint.

http://s35.photobucket.com/albums/d174/Brasempje/?action=view¤t=tensor.jpg

In equation 5 on the page that I linked above I do not see how I can get from term 3 to term 4. I end up with 3 kronecker delta's instead of just 1. Since there is a double index 'm' summation can be performed and it should lead to the same result as when you use a 'shortcut'. I can show what I do using 'latex' notation, with ^ = superscript, _ = subscript, d = kronecker delta, t = theta

'term 3' in Equation (5) reads:

(dt^i/dx^m) * (dx^m/dt^j)

in case I decide to do summation this equation will turn into:

(dt^i/dx^1) * (dx^1/dt^j) + (dt^i/dx^2) * (dx^2/dt^j) +

(dt^i/dx^3) * (dx^3/dt^j)

Each component of vector x is a function of all three components of vector theta. And each component of vector theta is a function of all three components of vector x. By the chain rule the last expression would become

dt^i/dt^j + dt^i/dt^j + dt^i/dt^j

this is:

d^i_j + d^i_j + d^i_j = 3 * d^i_j

so in case I decide to do the summation I end up with something different than I

would expect. 3 kronecker delta's instead of 1! Is there any objection to using a sum in this

case?

with kind regards,

Bremvil

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# Tensors (summation convention)

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