Help me to demonstrate and explain: Vrs=r∇s+s∇r and ∇.sv=(VS.v)+s(∇.v)

[mark_usc]

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Dear all:

Please help me to know how these expressions are obtained. Please recommendme some web reference to search more about the demonstrations.

Vrs=r∇s+s∇r

∇.sv=(VS.v)+s(∇.v)

This expressions are very important in transport phenomena and fluid mechanics.

With all the best

Marco Uscanga

 

[Fredrik]

Why did you write "V" in two places? Did you mean ∇? Are r and s scalar fields, and v a vector field? You should explain these things when you ask for help. Is "S" a typo?

$\nabla (rs)$ is a vector whose ith component is $\partial_i(rs)$. Use the product rule for derivatives.

$\nabla\cdot (sv)=\sum_i\partial_i(sv)_i =\sum_i\partial_i(sv_i)$. Now use the product rule again.

 

[mark_usc]

it seems good to me. Thanks a lot

Marco Uscanga