Calculating div(theta) and tangent curves

[renegade05]

Homework Statement

Calculate ∇Θ where Θ(x)=\frac{\vec{p} \cdot \vec{x}}{r^3}. Here \vec{p} is a constant vector and r=|\vec{x}|. In addition, sketch the tangent curves of the vector function ∇Θ for \vec{p}=p\hat{z}

(b) Calculate ∇ (cross) A → \vec{A}=\frac{\vec{m}x\vec{X}}{r^3} m is constant vector. Sketch the tangent curves of ∇(cross)A for \vec{m}=m\hat{z}

Homework Equations

gradient vector

The Attempt at a Solution

Well when I apply the gradient vector to the function Θ I get many terms and a very ugly answer. I am not sure if this will clean up nicely? Is there an easier way of doing this then brute force? Also, I am not sure how to represent the tangent curve of the vector function \vec{p}=p\hat{z} or \vec{m}=m\hat{z}.

 

[vanhees71]

Sometimes the Ricci calculus is easier than the nabla calculus. BTW: Both questions are not about div but about grad and curl.

For the first problem you have to calculate (Einstein summation convention implied)

\partial_j \Theta=\partial_j \left ( \frac{p_k x_k}{r^3} \right ).
This is now just the task to take the partial derivatives using the usual rules for differentiation (product rule in this case).

For the second problem note that
(\vec{\nabla} \times \vec{A})_j = \epsilon_{jkl} \partial_k A_l,
where \epsilon_{jkl} is the Levi-Civita symbol.