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grad(dg/dr) dot grad r = d^2 g/dr^2

Posted on 2008-11-01
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Last Modified: 2010-05-18
I'm trying to get from (13) to (14), here:
http://mathworld.wolfram.com/Laplacian.html

I've managed to show that the first terms equate:

dg/dr div(grad r) = 2/r dg/dr,

but I'm struggling to equate the second terms:

grad(dg/dr) dot grad r = d^2 g/dr^2

Can anyone spell it out for me please?

Thank you.
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Question by:InteractiveMind
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4 Comments
 
LVL 27

Expert Comment

by:aburr
ID: 22858227
The rs are unit vectors.    I do not think your starting equation is
grad(dg/dr) dot grad r
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LVL 25

Author Comment

by:InteractiveMind
ID: 22861982
The same equation can be derived if you assume that r is a function of (x,y,z) with r(x,y,z) = sqrt(x^2+y^2+z^2). In which case, grad r = (x,y,z)/||(x,y,z)||, which we can call r hat.
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Accepted Solution

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InteractiveMind earned 0 total points
ID: 22867134
<grad(dg/dr), grad r>
= (Drr g Dx r, Drr g Dy r, Drr g Dz r) dot grad r
= Drr g <grad r, grad r>
= Drr g

As required.
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