*assignment*Find Area/Volume from $d\vec{r}$*assignment*Homework##### Find Area/Volume from \(d\vec{r}\)

Static Fields 2023 (5 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.

- Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
- Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
- Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

*group*Vector Surface and Volume Elements*group*Small Group Activity30 min.

##### Vector Surface and Volume Elements

Static Fields 2023 (4 years)Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

*group*Scalar Surface and Volume Elements*group*Small Group Activity30 min.

##### Scalar Surface and Volume Elements

Static Fields 2023 (7 years)Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

*assignment*Cone Surface*assignment*Homework##### Cone Surface

Static Fields 2023 (6 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.

*assignment*Electric Field of a Finite Line*assignment*Homework##### Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

- Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
- Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

*assignment*Flux through a Plane*assignment*Homework##### Flux through a Plane

Static Fields 2023 (4 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).*assignment*Icecream Mass*assignment*Homework##### Icecream Mass

Static Fields 2023 (6 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*Memorize $d\vec{r}$*assignment*Homework##### Memorize \(d\vec{r}\)

Static Fields 2023 (3 years)Write \(\vec{dr}\) in rectangular, cylindrical, and spherical coordinates.

- Rectangular: \begin{equation} \vec{dr}= \end{equation}
- Cylindrical: \begin{equation} \vec{dr}= \end{equation}
- Spherical: \begin{equation} \vec{dr}= \end{equation}

*group*Vector Differential--Curvilinear*group*Small Group Activity30 min.

##### Vector Differential--Curvilinear

Vector Calculus II 23 (9 years)vector calculus coordinate systems curvilinear coordinates

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

*assignment*Electric Field from a Rod*assignment*Homework##### Electric Field from a Rod

Static Fields 2023 (5 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(xy\)-plane. The charge density \(\lambda\) is constant. Find the electric field at the point \((0,0,2L)\).-
Static Fields 2023 (4 years)
Find the surface area of a sphere
*using cylindrical coordinates*. Note: The fact that you can describe spheres nicely in cylindrical coordinates underlies the equal area cylindrical map project that allows you to draw maps of the earth where everything has the correct area, even if the shapes seem distorted. If you want to plot something like population density, you need an area preserving map projection.