## Cone Surface

• assignment Icecream Mass

assignment Homework

##### Icecream Mass
Static Fields 2022 (4 years)

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 Find Area/Volume from $d\vec{r}$

assignment Homework

##### Find Area/Volume from $d\vec{r}$
Static Fields 2022 (4 years)

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}

• group Flux through a Cone

group Small Group Activity

30 min.

##### Flux through a Cone
Static Fields 2022 (3 years)

Integration Sequence

Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .
• assignment Sphere in Cylindrical Coordinates

assignment Homework

##### Sphere in Cylindrical Coordinates
Static Fields 2022 (3 years) Find the surface area of a sphere using cylindrical coordinates.
• group Calculating Coefficients for a Power Series

group Small Group Activity

30 min.

##### Calculating Coefficients for a Power Series
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

$c_n={1\over n!}\, f^{(n)}(z_0)$

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a $\sin\theta$ (around both the origin and $\frac{\pi}{6}$). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

• assignment_ind Curvilinear Coordinates Introduction

assignment_ind Small White Board Question

10 min.

##### Curvilinear Coordinates Introduction
Static Fields 2022 (5 years)

Curvilinear Coordinate Sequence

First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles $\theta$ and $\phi$. Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
• groups Pineapples and Pumpkins

groups Whole Class Activity

10 min.

##### Pineapples and Pumpkins
Static Fields 2022 (4 years)

Integration Sequence

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element $d\tau$, interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

• face Statistical Analysis of Stern-Gerlach Experiments

face Lecture

30 min.

##### Statistical Analysis of Stern-Gerlach Experiments
• Static Fields 2022 (4 years)

• 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.