Differential Vector Elements

Reading: GEM § 1.2.2

1. (drVecMemorize)

   Give the expression for $d\vec{r}$ in rectangular, cylindrical, and spherical coordinates.

2. (DistanceCurvilinear)

   The distance $\left\vert\Vec r -\Vec r\,{}'\right\vert$ between the point $\Vec r\,{}'=(x\,{}',y\,{}',z\,{}')$ and the point $\Vec r=(x,y,z)$ is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

   1. Find the distance $\left\vert\Vec r -\Vec r\,{}'\right\vert$ between the point $\Vec r\,{}'=(x\,{}',y\,{}',z\,{}')$ and the point $\Vec r=(x,y,z)$ in rectangular coordinates.

   2. Show that this same distance written in cylindrical coordinates is: $$ \left|\Vec r -\Vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi\,{}'-\phi) +(z\,{}'-z)^2} $$

   3. Show that this same distance written in spherical coordinates is: $$ \left\vert\Vec r\,{}' -\Vec r\right\vert =\sqrt{r\,{}'^2+r^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi\,{}'-\phi) +\cos\theta\,{}'\cos\theta\right]} $$

   4. Now assume that $\Vec r\,{}'$ and $\Vec r$ are in the $x$-$y$ plane. Simplify the previous two formulas.

3. (Multipole–Challenge)

   Find the distance $\left\vert\rr -\rr'\right\vert$ between the point $\rr$ and the point $\rr'$ in terms of the magnitudes of $\rr$ and $\rr'$ and $\gamma$, the angle between them. (Do not choose a coordinate system.) Then assuming that $\rr»\rr'$, find a series expansion for $\left\vert\rr -\rr'\right\vert$, correct to fourth order. This expansion is the basis of multipole expansions, used in both electromagnetic theory and quantum mechanics.