tensor - Partial Differentiation

jg371 · **Posted:** Wed, 16 Apr 2008 21:43:01 UTC

I have undertaken to learn Tensor Calculus and have come across the following problem which I have some understanding of, however, I also have a couple of questions to clarify. Here goes with the problem:

If the $a_{i,j}$ are constants, calculate the partial derivative of $\frac{\partial}{\partial x_{k}}(a_{ij}x_{i}x_{j})$

The solution:

To begin with I did revert to $\sum{}$ notation:

$\sum_{ij}a_{ij}x_{i}x_{j}=\sum_{i\neq k, j\neq k}a_{ij}x_{i}x_{j}+\sum_{i=k, j\neq k}a_{ij}x_{i}x_{j}+\sum_{i\neq k, j=k}a_{ij}x_{i}x_{j}+\sum_{i=k, j=k}a_{ij}x_{i}x_{j}$

$\sum_{ij}a_{ij}x_{i}x_{j}=C +\left(\sum_{j\neq k}a_{kj}x_{j}\right)x_{k}+\left(\sum_{i\neq k}a_{ik}x_{i}\right)x_{k}+a_{kk}(x_{k})^2$

where C is independent of $x_{k}$

Differentiating with respect to $x_{k}$ :

$\frac{\partial }{\partial x_{k}}\left(\sum_{ij}a_{ij}x_{i}x_{j}\right)= 0 +\sum_{j\neq k}a_{kj}x_{j}+\sum_{i\neq k}a_{ik}x_{i}+2a_{kk}x_{k}$

$\frac{\partial }{\partial x_{k}}\left(\sum_{ij}a_{ij}x_{i}x_{j}\right)= \sum_{j}a_{kj}x_{j}+\sum_{i}a_{ik}x_{i}$

Reverting to Einstein's summation convention:

$\frac{\partial }{\partial x_{k}}\left(\sum_{ij}a_{ij}x_{i}x_{j}\right)=a_{kj}x_{j}+a_{ik}x_{i}= a_{ki}x_{i}+a_ik}x_{i}=\left(a_{ki}+a_{ik)\right)x_{i}$

Now, here are my questions:

Why does $\sum_{i\neq k, j\neq k}a_{ij}x_{i}x_{j}$ = C$

and why does the term $2a_{kk}x_{k}$ vanishes$

Thank you kindly for your kind assistance

jg

commutative · **Joined:** Sat, 18 Mar 2006 08:42:24 UTC **Posts:** 834

jg371 wrote:

Now, here are my questions:

Why does $\sum_{i\neq k, j\neq k}a_{ij}x_{i}x_{j}$ = C$

here means "constant with respect to" x_k.

there's no x_k

in the above sum.

and why does the term $2a_{kk}x_{k}$ vanishes$

it doesn't vanish:

$\sum_{j\neq k}a_{kj}x_{j}+\sum_{i\neq k}a_{ik}x_{i}+2a_{kk}x_{k}$

$=(\sum_{j\neq k}a_{kj}x_{j}+ a_{kk}x_k) + (\sum_{i\neq k}a_{ik}x_{i}+ a_{kk}x_k)$

$=\sum_j a_{kj}x_j + \sum_i a_{ik}x_i.$ note that now the sums are over all j and i not only over $j \neq k$ and $i \neq k.$

S.O.S. Mathematics CyberBoard

tensor - Partial Differentiation

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