assignment Homework

Polar vs. Spherical Coordinates

1. The direction of \(\theta=0\) in spherical coordinates is the same as the direction of out of the plane in plane polar coordinates.
2. Given the correspondance above, then if we choose the \(\theta\) of spherical coordinates is to be \(\pi/2\), we restrict to the equatorial plane of spherical coordinates.

assignment Homework

Memorize \(d\vec{r}\)

Write \(\vec{dr}\) in rectangular, cylindrical, and spherical coordinates.

1. Rectangular: \begin{equation} \vec{dr}= \end{equation}
2. Cylindrical: \begin{equation} \vec{dr}= \end{equation}
3. Spherical: \begin{equation} \vec{dr}= \end{equation}

assignment Homework

Icecream Mass

Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

assignment Homework

The Gradient for a Point Charge

Gradient Sequence

The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}

1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

assignment Homework

Cone Surface

• Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
• Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

assignment Homework

Find Area/Volume from \(d\vec{r}\)

Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

assignment Homework

Distance Formula in Curvilinear Coordinates

Ring Cycle Sequence

The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) 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 \(\vec r\) and the point \(\vec r'\) in rectangular coordinates.

   

2. Show that this same distance written in cylindrical coordinates is: \begin{equation} \left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2} \end{equation}
3. Show that this same distance written in spherical coordinates is: \begin{equation} \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]} \end{equation}
4. Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify the previous two formulas.

assignment Homework

Sphere in Cylindrical Coordinates