My physics teacher assigned this problem: “Evaluate →∇→x2\vec\nabla \vec x^2 where →x2=x2+y2+z2\vec x^2 = x^2 + y^2 +z^2 and →∇=ˆı∂∂x+ˆȷ∂∂y+ˆk∂∂z\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 →x2=x2+y2+z2\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, ˆı,ˆȷ,ˆk\hat{\imath},\hat{\jmath},\hat{k}.

Could anyone shed some light on the subject? What is this notation?

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2 Answers

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The notation →x2=x2+y2+z2\vec x ^2 = x^2 + y^2 +z^2 seems to be the square of the length of the vector →x\vec x. This is often written as ‖→x‖2\|\vec x \|^2, where \|\vec x\|\|\vec x\| is the length of the vector \vec x\vec x.

The symbol \vec\nabla\vec\nabla is the gradient operator. We can think of \vec x^2\vec x^2 as a function where

(x,y,z) \longmapsto x^2 + y^2 + z^2(x,y,z) \longmapsto x^2 + y^2 + z^2

The gradient of the function \vec x^2\vec x^2 would give

\vec \nabla \vec x^2 = 2x{\bf i} \ + \ 2y{\bf j} \ + \ 2z{\bf k} \vec \nabla \vec x^2 = 2x{\bf i} \ + \ 2y{\bf j} \ + \ 2z{\bf k}

The notation \vec{x}^2\vec{x}^2 is presumably being used to mean (\vec{x})^2(\vec{x})^2, which is equivalent to \vec{x}\cdot\vec{x}\vec{x}\cdot\vec{x}. It’s not a vector, it’s just the square of a norm.