Vector calculus identities - is this right?

[quarky2001]

I'm manipulating an equation, and I think I am correct in doing this, but not sure. Could someone tell me if the equality I've written below is true?

$$[\nabla\cdot [\rho\vec{v}\vec{v}] ]\cdot\vec{v} = \nabla\cdot[\frac{1}{2}\rho v^2 \vec{v}]$$

(where $$\rho$$ is dependent on position)

*NOTE* that my use of $$\vec{v}\vec{v}$$ (beside each other without a dot or cross) denotes a tensor quantity, i.e. $$\nabla\cdot(\rho\vec{v}\vec{v}) = \rho(\vec{v}\cdot\nabla)\vec{v} + \vec{v}\nabla\cdot(\rho\vec{v})$$.

 

[lanedance]

so you mean an outer product
http://en.wikipedia.org/wiki/Outer_product

now i assume v = v(x,y,z). also is ro a scalar constant, or a scalar function

something like this is probably simpler in index notation, though you'll have to be careful with grad, this work in progress but hopefully helpful (and not rigorous)

so first let's define the dot product of the grad operator as
$$\nabla\cdot \vec{v} = \frac{\partial }{\partial x_i} \delta_{ij} v_j = \frac{\partial }{\partial x_i} v_i$$

for the outer product (becoming in effect a 3x3 matrix)
$$\rho \vec{v} \otimes \vec{v} = v_j v_k$$

then extend to (and this is a bit of a strecth)
$$\nabla\cdot ( \rho \vec{v} \otimes \vec{v}) = \frac{\partial }{\partial x_i} \delta_{ij} \rho v_j v_k$$

see if you can extend it from there (note I've only used index summation notation and haven't gone near contra/covariance & invariance and tensor tranformations, which will become important when you start referring to tensors...)