How to find the divergence and the curl of the given vectors?
a. $( \vec{u} \cdot \vec{r}) \vec{v}$
b. $( \vec{u} \cdot \vec{r}) \vec{r}$
c. $( \vec{u} \times \vec{r})$
d. $ \vec{r} \times(\vec{u} \times \vec{r})$
e. $ \psi (r) (\vec{u} \times \vec{r})$
where $\vec{u}$ and $\vec{v}$ are the constant vectors, $\vec{r}$ is the radius vector and $\psi(r)$ is a scalar function of the magnitude r of the $\vec{r}$
Thanks.
Here is one example.
Using the product rule for differentiation on Part e, we have
$$\nabla \cdot (\psi(\vec r)(\vec u \times \vec r))=(\vec u \times \vec r)\cdot \nabla \psi(\vec r)+\psi(\vec r)\nabla \cdot (\vec u \times \vec r)$$
The first term can be expressed alternatively as
$$(\vec u \times \vec r)\cdot \nabla \psi(\vec r)=\vec u\cdot (\vec r \times \nabla \psi(\vec r))$$
For the divergence component of the second term, we have (using summation notation)
$$\begin{align} \nabla \cdot (\vec u \times \vec r)&=\partial_i \hat x_i \cdot (u_j \hat x_j \times x_k\hat x_k)\\\\ &=u_j \hat x_i \cdot ( \hat x_j \times \hat x_k)\partial_i (x_k)\\\\ &=u_j \hat x_i \cdot ( \hat x_j \times \hat x_k)\delta_{ik} \\\\ &=u_j \hat x_i \cdot ( \hat x_j \times \hat x_i)\\\\ &=0 \end{align}$$
Thus,
$$\nabla \cdot (\psi(\vec r)(\vec u \times \vec r))=\vec u\cdot (\vec r \times \nabla \psi(\vec r))$$