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

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In summary, the conversation is about someone seeking help to understand the expressions Vrs=r∇s+s∇r and ∇.sv=(VS.v)+s(∇.v), which are important in transport phenomena and fluid mechanics. The person also asks for recommendations for web references to learn more about the demonstrations. The expert provides a summary of how to obtain the expressions using the product rule for derivatives.
  • #1
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
 
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  • #2
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.
 
  • #3
it seems good to me. Thanks a lot

Marco Uscanga
 

1. What does the equation Vrs=r∇s+s∇r represent?

The first equation, Vrs=r∇s+s∇r, is known as the vector product rule and represents the cross product between two vectors, r and s.

2. How is the cross product calculated using Vrs=r∇s+s∇r?

The cross product between two vectors, r and s, is calculated by taking the product of the first vector, r, with the gradient of the second vector, ∇s, and adding it to the product of the second vector, s, with the gradient of the first vector, ∇r.

3. Can you give an example of how to use Vrs=r∇s+s∇r?

One example of using the vector product rule is in calculating the torque, or turning force, on a rotating object. The torque vector can be found by taking the cross product of the position vector from the axis of rotation to a point on the object, with the force vector acting on that point.

4. What does the equation ∇.sv=(VS.v)+s(∇.v) represent?

The second equation, ∇.sv=(VS.v)+s(∇.v), is known as the dot product rule and represents the dot product between a scalar, s, and a vector, v.

5. How is the dot product calculated using ∇.sv=(VS.v)+s(∇.v)?

The dot product between a scalar, s, and a vector, v, is calculated by taking the product of the scalar, s, with the divergence of the vector, ∇.v, and adding it to the product of the vector, v, with the scalar gradient, VS.

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