## Differentials of Two Variables

• assignment Flux through a Plane

assignment Homework

##### Flux through a Plane
Static Fields 2022 (3 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 Differentials of One Variable

assignment Homework

##### Differentials of One Variable
Static Fields 2022 (4 years) Find the total differential of the following functions:
1. $y=3x^2 + 4\cos 2x$
2. $y=3x^2\cos kx$ (where $k$ is a constant)
3. $y=\frac{\cos 7x}{x^2}$
4. $y=\cos(3x^2-2)$
• assignment Rubber Sheet

assignment Homework

##### Rubber Sheet
Energy and Entropy 2021 (2 years)

Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call $y$, and the distance of horizontal stretch we will call $x$.

If I pull the bottom down by a small distance $\Delta y$, with no horizontal force, what is the resulting change in width $\Delta x$? Express your answer in terms of partial derivatives of the potential energy $U(x,y)$.

• assignment The puddle

assignment Homework

##### The puddle
differentials Static Fields 2022 (3 years) The depth of a puddle in millimeters is given by $h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)$ Your path through the puddle is given by $x=3t \qquad y=4t$ and your current position is $x=3$, $y=4$, with $x$ and $y$ also in millimeters, and $t$ in seconds.
1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
3. FOOD FOR THOUGHT (optional)
There is a walkway over the puddle at $x=10$. At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
• 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}

• format_list_numbered Integration Sequence

format_list_numbered Sequence

##### Integration Sequence
Students learn/review how to do integrals in a multivariable context, using the vector differential $d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}$ and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.
• assignment Cube Charge

assignment Homework

##### Cube Charge
charge density

Integration Sequence

Static Fields 2022 (4 years)
1. Charge is distributed throughout the volume of a dielectric cube with charge density $\rho=\beta z^2$, where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
2. In a new physical situation: Charge is distributed on the surface of a cube with charge density $\sigma=\alpha z$ where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge on the cube? Don't forget about the top and bottom of the cube.
• assignment Memorize $d\vec{r}$

assignment Homework

##### Memorize $d\vec{r}$
Static Fields 2022 (2 years)

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

1. Rectangular: $$\vec{dr}=$$
2. Cylindrical: $$\vec{dr}=$$
3. Spherical: $$\vec{dr}=$$

• group Scalar Surface and Volume Elements

group Small Group Activity

30 min.

##### Scalar Surface and Volume Elements
Static Fields 2022 (4 years)

Integration Sequence

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_ind Vector Differential--Rectangular

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
Static Fields 2022 (7 years)

Integration Sequence

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

• Static Fields 2022 (5 years) Find the total differential of the following functions:
1. $y=3u^2 + 4\cos 3v$
2. $y=3uv$
3. $y=3u^2\cos wv$
4. $y=u\cos(3v^2-2)$