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Incand
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Homework Statement
Show that ##\nabla u_i \cdot \frac{\partial \vec r}{\partial u_i} = \delta_{ij}##.
(##u_i## is assumed to be a generalized coordinate.)
Homework Equations
Gradient in curvilinear coordinates
##\nabla \phi = \sum_{i=1}^3 \vec e_i \frac{1}{h_i} \frac{\partial \phi}{\partial u_i}##
The Attempt at a Solution
So what I need to show is that the gradient is orthogonal to the (other two) partial derivatives (but it doesn't have to be parallel to the third since the system doesn't have to be orthogonal!).
The expression can then be written as
##\sum_{k=1}^3 \left( \vec e_k \frac{1}{h_k} \frac{\partial u_i}{\partial u_k} \right) \cdot \frac{\partial \vec r}{\partial u_j} = \sum_{k=1}^3 \frac{1}{h_k}\frac{\partial u_i}{\partial u_k}\frac{\partial r_k}{\partial u_j} ##.
I'm thinking I could use the chain rule here but I don't seem to get anywhere. Another thing of note is that ##\frac{1}{h_k}\frac{\partial \vec r}{\partial u_k} = \vec e_k## which may be of use somewhere.