I'm having some trouble wrapping my mind around some vector problems. I'm hoping you could please help me to clarify it.
Let's say $\vec{r}=(x_{1},x_{2},…,x_{n})$, $\mathbf r = |\vec{r}|$. How should I correctly interpret $\mathbf r$?
In addition, I am looking for a way to compute $\nabla \cdot \frac{\vec{r}}{\mathbf r^k}$. Could you please walk me through how to do it?
Thank you so much in advance!
First, $\bf r$, which is $|r|$, should be interpreted as the length of the vector $r$. So say we have a vector in $\Bbb R^{2}$. Let's pick $(1,2)$. Then $|(1,2)|$ means $\sqrt{(1 - 0)^{2} + (2-0)^{2}}$, which is the distance of the line with one endpoint the origin $(0,0)$ and the other endpoint $(1,2)$. For every point in $\Bbb R^{2}$, you can draw a line connecting $(0,0)$ and that point. The absolute value of the point means the length of this line. If instead of points you are considering $(1,2)$ as a vector, then $|(1,2)|$ can be interpreted as the "magnitude" or length of this vector (since vectors always start from the origin $(0,0)$).
Now, if I have a vector like $v = (2x_{1}, 3x_{2} - x_{2}^{2}, \sin(x_{3}) + 2x_{1})$, then $\nabla \cdot v$ means you apply the $\frac{\partial}{\partial x_{1}}$ operator to the first entry, PLUS the $\frac{\partial}{\partial x_{2}}$ operator applied to the second entry, PLUS the $\frac{\partial}{\partial x_{3}}$ applied to the third entry.
So I have \begin{split}\nabla \cdot v &= \frac{\partial}{\partial x_{1}}(2x_{1}) + \frac{\partial}{\partial x_{2}}(3x_{2} - x_{2}^{2}) + \frac{\partial}{\partial x_{3}}(\sin(x_{3}) + 2x_{1}) \\ &= 2 + 3 - 2x_{2} + \cos(x_{3}) \\ &= 5 - 2x_{2} + \cos(x_{3}). \end{split}