My physics teacher assigned this problem: "Evaluate $\vec\nabla \vec x^2 $ where $\vec x^2 = x^2 + y^2 +z^2$ and $\vec\nabla = \hat{\imath}\frac{\partial}{\partial x}+\hat{\jmath}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z}$.
I know how to compute the problem. But I am confused about this vector $\vec x^2 = x^2 + y^2 +z^2$. I thought vectors have a norm and direction. But it looks like this is only a norm. I am used to vectors always having components, $\hat{\imath},\hat{\jmath},\hat{k}$.
Could anyone shed some light on the subject? What is this notation?
2 Answers
The notation $\vec x ^2 = x^2 + y^2 +z^2$ seems to be the square of the length of the vector $\vec x$. This is often written as $\|\vec x \|^2$, where $\|\vec x\|$ is the length of the vector $\vec x$.
The symbol $\vec\nabla$ is the gradient operator. We can think of $\vec x^2$ as a function where $$(x,y,z) \longmapsto x^2 + y^2 + z^2$$ The gradient of the function $\vec x^2$ would give $$\vec \nabla \vec x^2 = 2x{\bf i} \ + \ 2y{\bf j} \ + \ 2z{\bf k} $$
The notation $\vec{x}^2$ is presumably being used to mean $(\vec{x})^2$, which is equivalent to $\vec{x}\cdot\vec{x}$. It's not a vector, it's just the square of a norm.
