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# Derivatives of Tensors

Hi all,

I am looking for a general definition of the derivative of a tenorial product (e.g. when the expression for Stress contains nonlinear terms in deformation gradient, **F** ).

I start with a very simple example:

∂F_pq/∂F_mn = δ_pm δ_qn , i.e. Kronecker delta with first index of F_pq and first index of F_mn, and second Kronecker delta for second pair of indices q & n.

However the problem arises when we have a product of two or more tensors. Is the following (using chain rule) a valid derivative?

∂ (A_ij B_jk) /∂F_mn = [∂A_ij / ∂F_mn ] B_jk + A_ij [∂B_jk / ∂F_mn]

I used this method of using indicies, and found that results are sometimes very different than what is written (with any explaination) in textbooks.

E.g. if A=F^-1 and B=F then

∂(F^-1_ij F_jk) /∂F_mn = [∂F^-1_ij / ∂F_mn ] F_jk + F^-1_ij [∂F_jk / ∂F_mn] = 0 (because **F**-1 • **F** = **I** and ∂(**I**)/∂**F** = **0**

Therefore, [∂F^-1_ij / ∂F_mn ] F_jk = - F^-1_ij [∂F_jk / ∂F_mn]

and [∂F^-1_ij / ∂F_mn ] = - F^-1_ij δ_jm δ_kn F^-1_jk

hence [∂F-1_ij / ∂F_mn ] = - F-1_im F-1_jn

However, in most books, Matrix Cook Book, and the Wikipedia article for Tensor derivatives, I found the second term in result is different (the indices have interchanged their positions), i.e.

[∂F^-1_ij / ∂F_mn ] = - F^-1_im F-^1_nj

I will be very thankful if anyone can kindly explain what went wrong in the derivative (above in red) which I have calculated

Best regards,

Mubeen

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## Comments

## Take a look at this link, i.e

Take a look at this link, i.e. Equation (1.15.18).

http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks...

## Thanks!

Thanks for pointing to Kelly's notes. I read these notes last year, but missed this part.

## 'Therefore, [∂F^-1_ij / ∂F_mn

'Therefore, [∂F^-1_ij / ∂F_mn ] F_jk = - F^-1_ij [∂F_jk / ∂F_mn]'. The derivation up to this line is correct.

In the next step. Multiply both sides of the equation by F^-1_kp instead of F^-1_jk. Note that F_jkF^-1_kp=δ_jp.

In your derivation, F_jkF^-1_jk is mistakenly taken to be 1, which makes the result wrong.

## Tons of Thanks!!!

Your explaination is very clear, I must say.