vector calculus formulas proof

Question

I would like to ask how to prove these two formulas.

for T is a scalar and r is radial component of a vector in spherical coordinate.

(1)

$$ {\bf{r}\cdot[(r\cdot\nabla)\nabla}T+2\nabla T]={\bf{r}\cdot\nabla} T+{\bf r\cdot\nabla(r\cdot\nabla} T) $$

(2) $$ {\nabla^2(T\bf{r})}={\bf {r}}\nabla^2T+T\nabla^2{\bf {r}}+2\nabla T\cdot\nabla {\bf {r}} $$

Thank you !!

Answer 1

For (1), note that \begin{align*} \mathbf{r}\cdot[(\mathbf{r}\cdot\nabla)\nabla T+2\nabla T] &=x_i[(x_j\partial_j)\partial_i T+2\partial_i T]\\ &=x_i(x_j\partial_j)\partial_i T+2x_i\partial_i T\\ &=x_j\partial_jx_i\partial_i T-x_j(\partial_j x_i)\partial_i T+2x_i\partial_i T\\ &=x_j\partial_jx_i\partial_i T-x_j\delta_{ij}\partial_i T+2x_i\partial_i T\\ &=x_j\partial_jx_i\partial_i T-x_i\partial_i T+2x_i\partial_i T\\ &=x_j\partial_jx_i\partial_i T+x_i\partial_i T\\ &=\mathbf{r}\cdot\nabla(\mathbf{r}\cdot\nabla T)+\mathbf{r}\cdot\nabla T \end{align*}

(2) is similar.