Using the Divergence Theorem to Solve Vector Calculus Problems

[SP90]

Homework Statement

Homework Equations

So I have that $v \otimes n = \left( \begin{array}{ccc} v_{1}n_{1} & v_{1}n_{2} & v_{1}n_{3} \\ v_{2}n_{1} & v_{2}n_{2} & v_{2}n_{3} \\ v_{3}n_{1} & v_{3}n_{2} & v_{3}n_{3} \end{array} \right)$

The Attempt at a Solution

I've tried applying the Divergence theorem for Tensors:
$\int_{\partial B} ( v \otimes n )n dA = \int_{B} \nabla \cdot ( v \otimes n ) dV$

But that doesn't lead anywhere particularly useful. I thought it might be worth noting that $\nabla \cdot ( v \otimes n ) = \frac{dv_{1}}{dx_{1}}n_{1}+\frac{dv_{2}}{dx_{2}}n_{2} + \frac{dv_{3}}{dx_{3}}n_{3}$ but I can't seem to get anywhere near $\nabla v$

And this problem isn't homework, it's just an optional exercise, but it's frustrated me for a while and I figured I should get some pointers.

 

[clamtrox]

I thought it might be worth noting that $\nabla \cdot ( v \otimes n ) = \frac{dv_{1}}{dx_{1}}n_{1}+\frac{dv_{2}}{dx_{2}}n_{2} + \frac{dv_{3}}{dx_{3}}n_{3}$ but I can't seem to get anywhere near $\nabla v$

No it's not! I think you will find (assuming n is constant) that $\nabla \cdot ( v \otimes n ) = n \cdot (\nabla v) + n (\nabla \cdot v)$

Especially note that it is a vector and not a scalar

 

[SP90]

Isn't $\nabla \cdot v \otimes n = (n_{1}(\frac{dv_{1}}{dx_{1}}+\frac{dv_{2}}{dx_{2}}+\frac{dv_{3}}{dx_{3}}), n_{2}(\frac{dv_{1}}{dx_{1}}+\frac{dv_{2}}{dx_{2}}+\frac{dv_{3}}{dx_{3}}), n_{3}(\frac{dv_{1}}{dx_{1}}+\frac{dv_{2}}{dx_{2}}+\frac{dv_{3}}{dx_{3}}))$

Which is $n \cdot \nabla v$?

This would make sense since it gives that

$\int_{\partial B} ( v \otimes n )n dA = \int_{B} (\nabla v) ndV$

And then since n is just so constant vector, the result follows.

Is that right? Or am I missing something?

 

[clamtrox]

Isn't $\nabla \cdot v \otimes n = (n_{1}(\frac{dv_{1}}{dx_{1}}+\frac{dv_{2}}{dx_{2}}+\frac{dv_{3}}{dx_{3}}), n_{2}(\frac{dv_{1}}{dx_{1}}+\frac{dv_{2}}{dx_{2}}+\frac{dv_{3}}{dx_{3}}), n_{3}(\frac{dv_{1}}{dx_{1}}+\frac{dv_{2}}{dx_{2}}+\frac{dv_{3}}{dx_{3}}))$

Which is $n \cdot \nabla v$?

This would make sense since it gives that

$\int_{\partial B} ( v \otimes n )n dA = \int_{B} (\nabla v) ndV$

And then since n is just so constant vector, the result follows.

Is that right? Or am I missing something?

Of course not; you can't just get rid of the terms that give you trouble... This is what you need to show, but the formula I gave above is still correct.