Understanding Index Notation and Tensor Operations in Vector Calculus

[Niles]

Homework Statement

Hi

I have a vector v. According to my book, the following is valid:
$$\frac{1}{2}\nabla v^2-v\cdot \nabla v = v\times \nabla \times v$$
I disagree with this, because the first term on the LHS I can write as (partial differentiation)
$$\frac{1}{2}\partial_i v_jv_j = v_j\partial_i v_j$$
which is just $v\cdot \nabla v$. So IMO it should equal 0.

What is wrong here?

 

[pasmith]

Homework Statement

Hi

I have a vector v. According to my book, the following is valid:
$$\frac{1}{2}\nabla v^2-v\cdot \nabla v = v\times \nabla \times v$$

I disagree with this, because the first term on the LHS I can write as (partial differentiation)
$$\frac{1}{2}\partial_i v_jv_j = v_j\partial_i v_j$$
which is just $v\cdot \nabla v$. So IMO it should equal 0.

What is wrong here?

$$v_j \partial_i v_j \neq v_j \partial_j v_i = (v \cdot \nabla) v_i$$

Note that on the left the index on the derivative is i (a free index), but on the right the index on the derivative is j (a dummy index).

 

[Niles]

Thanks. I hope there is omething else you can help me with (you seem to have experience with this kind of notation). Is the following always true for some vector u?

$$\nabla \cdot (\nabla u) = \nabla (\nabla \cdot u)$$