I would like to ask how to prove these two formulas. for T is a scalar and r is radial component of a vector in spherical coordinate. (1) r *[(r * grad)grad T+2grad T]=r * grad T+r * grad(r * grad T) (2) grad^2(Tr)=r grad^2 T+T grad^2 r+2 grad T * grad r

muroscamsey

muroscamsey

Answered question

2022-08-12

I would like to ask how to prove these two formulas.
for T is a scalar and r is radial component of a vector in spherical coordinate.
(1) r [ ( r ) T + 2 T ] = r T + r ( r T )
(2) 2 ( T r ) = r 2 T + T 2 r + 2 T r

Answer & Explanation

wietselau

wietselau

Beginner2022-08-13Added 28 answers

For (1), note that
r [ ( r ) T + 2 T ] = x i [ ( x j j ) i T + 2 i T ] = x i ( x j j ) i T + 2 x i i T = x j j x i i T x j ( j x i ) i T + 2 x i i T = x j j x i i T x j δ i j i T + 2 x i i T = x j j x i i T x i i T + 2 x i i T = x j j x i i T + x i i T = r ( r T ) + r T
(2) is similar.

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