Partial Differentiation with Indicial Notation (Ritz Method for FEM)

[bugatti79]

Folks,

I am stuck on an example which is partial differenting a functional with indicial notation

The functional $\displaystyle I(c_1,c_2,...c_N)=\frac{1}{2} \int_0^1 \left [ \left (\sum\limits_{j=1}^N c_j \frac{d \phi_j}{dx}\right )^2-\left(\sum\limits_{j=1}^N c_j \phi_j\right)^2+2x^2 \left(\sum\limits_{j=1}^N c_j \phi_j \right)\right ]dx$

Differentiating this wrt to $c_i$ ie

$\displaystyle \frac{\partial I}{\partial c_i}= \int_0^1 \left[ \frac{d \phi_i}{dx} \left(\sum\limits_{j=1}^N c_j \frac{ d\phi_j}{dx} \right )-\phi_i \left(\sum\limits_{j=1}^N c_j \phi_j \right) + \phi_i x^2 \right] dx$

I don't understand how this last line is obtained. If we focus on the first term. I realize that there is a chain rule procedure. My attempt on the first term in first eqn was

$\displaystyle \frac{\partial I}{\partial c_i}= \frac{1}{2}\int_0^1 \left[ 2\left(\sum\limits_{j=1}^N c_j \frac{ d\phi_j}{dx} \right )\frac{d}{dc_i}\left(\sum\limits_{j=1}^N c_j\frac{ d\phi_j}{dx}\right) \right]dx$

Not sure how to handle the indicial notation or how proceed any further.
Any help will be greatly appreciated...thanks

 

[voko]

$\displaystyle \frac{\partial I}{\partial c_i}= \frac{1}{2}\int_0^1 \left[ 2\left(\sum\limits_{j=1}^N c_j \frac{ d\phi_j}{dx} \right )\frac{d}{dc_i}\left(\sum\limits_{j=1}^N c_j\frac{ d\phi_j}{dx}\right) \right]dx$

\frac{d}{dc_i}\left(\sum\limits_{j=1}^N c_j\frac{ d\phi_j}{dx}\right) = \frac{d}{dc_i}\left(c_1\frac{ d\phi_1}{dx} + ... + c_i\frac{ d\phi_i}{dx} + ... + c_N\frac{ d\phi_N}{dx}\right) = 0 + ... + \frac{ d\phi_i}{dx} + ... 0

 

[bugatti79]

\frac{d}{dc_i}\left(\sum\limits_{j=1}^N c_j\frac{ d\phi_j}{dx}\right) = \frac{d}{dc_i}\left(c_1\frac{ d\phi_1}{dx} + ... + c_i\frac{ d\phi_i}{dx} + ... + c_N\frac{ d\phi_N}{dx}\right) = 0 + ... + \frac{ d\phi_i}{dx} + ... 0

Thank you so much. So the i is some integer between j=1 and j=N.

Cheers