Confusing index notation involving grad of w cross r

[troytroy]

Homework Statement

consider the position vector expressed in terms of its cartesian components, r=xiei. Let w=wjej be a fixed vector whose components wj are constants that do not depend on the xi, so that δwj/δxi = 0

Homework Equations

I am trying to evaluate ∇((wXr)^2)

The Attempt at a Solution

 

[gabbagabbahey]

Homework Statement

consider the position vector expressed in terms of its cartesian components, r=xiei. Let w=wjej be a fixed vector whose components wj are constants that do not depend on the xi, so that δwj/δxi = 0

Homework Equations

I am trying to evaluate ∇((wXr)^2)

The Attempt at a Solution

Hi troytroy, welcome to PF!

What have you tried and where are you stuck?

 

[troytroy]

I am getting confused on where to begin when using index notation for these kind of problems

 

[gabbagabbahey]

I am getting confused on where to begin when using index notation for these kind of problems

Well, here you are being asked to calulate the gradient of some scalar function, so a good place to start would be to look up the expression for the gradient of a general scalar function $f$ in index notation. What is that?

Next consider that in this case, the scalar function in question is the scalar product of of a vector with itself, $(\mathbf{w}\times\mathbf{r})^2$ (the norm-squared of a vector is usually written as $||\mathbf{v}||^2$, but some authors will use more clumsy notation and just call it $\mathbf{v}^2$. Either way the norm-squared of a vector is given by the scalar product of a vector with itself). So, how do you express the scalar product of a vector with itself in index notation?

Finally, consider that the vector whose norm-square you are taking the gradient of is, in this case, the cross product of a vector with another vector, $\mathbf{w}\times\mathbf{r}$. How do you represent a cross product like this in index notation?