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 [itex]f[/itex] 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, [itex](\mathbf{w}\times\mathbf{r})^2[/itex] (the norm-squared of a vector is usually written as [itex]||\mathbf{v}||^2[/itex], but some authors will use more clumsy notation and just call it [itex]\mathbf{v}^2[/itex]. 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, [itex]\mathbf{w}\times\mathbf{r}[/itex]. How do you represent a cross product like this in index notation?

 

[paranormal]

The notation ∇((wXr)^2) can be rewritten as ∇(wXr)·∇(wXr), where ∇(wXr) represents the gradient of the vector wXr. Using the chain rule, we can expand this as ∇(wXr)·(∇wXr·∇r + wX∇∇r), where ∇w and ∇r represent the gradients of the vectors w and r, respectively. Since w is a fixed vector with constant components, its gradient is equal to zero, and thus the term ∇wXr becomes zero. This leaves us with ∇(wXr)·wX∇∇r. We can further simplify this by using the identity ∇(wXr) = rX∇w + wX∇r, giving us (rX∇w + wX∇r)·wX∇∇r. This can be expanded as (r·w)(∇w·∇∇r) + (w·w)(∇r·∇∇r). Since w is a fixed vector, its dot product with itself is equal to its magnitude squared, making the second term in the expression equal to ∥w∥^2(∇r·∇∇r). The first term, (r·w)(∇w·∇∇r), can be rewritten as (r·w)(∇w·∇r)·∇r, using the identity ∇∇r = ∇r. Finally, we can substitute back in the original notation to get the expression (r·w)(∇w·∇r)·∇r + ∥w∥^2(∇r·∇r). This may seem like a confusing notation at first, but by breaking it down and using identities, we can simplify it and better understand the meaning behind it.