[ Follow Ups ] [ Post Followup ] [ Calculus Message Board ] [ FAQ ]

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:

V'i=aijVj

Note: I'm using the Einstein summation notation where summation is
over repeated indices. In this case the j on the right hand side.

where
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:
(x1',x2',x3')
in one system and be :
(x1,x2,x3)
in the other. Therefore:

f'(x1',x2',x3')=f(x1,x2,x3)

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:

Del f=i(df)/(dx)+j(df)/(dy)+k(df)/(dz)

Note: all d's are partial derivatives.

Name:
E-Mail:

Subject: