Posted by Brad Paul on October 07, 2002 at 18:46:21:
In Reply to: Gradient function posted by Cathy on October 07, 2002 at 14:48:03:
: For my GCSE maths (higher tier) i have to do coursework on the gradient function. Im doing fine... until I got to the part where you have to prove the gradient function- i really dont get this part. Does anybody understand?
A vector transformation can be written as:
Note: I'm using the Einstein summation notation where summation is
over repeated indices. In this case the j on the right hand side.
aij=(d xj)/(d xi')
A scalar function has the same value at a given point in space
regardless of the coordinate system. Let this point be:
in one system and be :
in the other. Therefore:
By differentiating with respect to xi' we may write:
(d f')/(d xi')=
(d xj)/(d xi')(d f)/(d xj)
We then notice that this is a definition of a vector transformation.
In Cartesian coordinates this is just:
Note: all d's are partial derivatives.
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