None

##### Circle Trig Complex

Find the rectangular coordinates of the point where the angle $\frac{5\pi}{3}$ meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

• Found in: Quantum Fundamentals course(s)

None

##### Circle Trigonometry

On the following diagrams, mark both $\theta$ and $\sin\theta$ for $\theta_1=\frac{5\pi}{6}$ and $\theta_2=\frac{7\pi}{6}$. Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)

• Found in: Quantum Fundamentals course(s)

group Small Group Activity

5 min.

##### Maxima and Minima
This small group activity introduces students to constrained optimization problems. Students work in small groups to optimize a simple function on a given region. The whole class wrap-up discussion emphasizes the importance of the boundary.
• Found in: Vector Calculus I course(s)

group Small Group Activity

30 min.

##### The Cylinder
This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.
• Found in: Vector Calculus I course(s)

None

##### Graphs Involving the Distance Formula

Learn more about the geometry of $\vert \vec{r}-\vec{r'}\vert$ in two dimensions.

1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
2. Make a sketch of the graph $$\vert \vec{r} - \vec{a} \vert = 2$$

for each of the following values of $\vec a$: \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

3. Derive a more familiar equation equivalent to $$\vert \vec r - \vec a \vert = 2$$ for arbitrary $\vec a$, by expanding $\vec r$ and $\vec a$ in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
4. Write a brief description of the geometric meaning of the equation $$\vert \vec r - \vec a \vert = 2$$

• Found in: Static Fields, AIMS Maxwell course(s)

group Small Group Activity

30 min.

##### Acceleration
• Found in: Vector Calculus II course(s)

groups Whole Class Activity

10 min.

##### Pineapples and Pumpkins

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 and volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distributed 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.

• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop course(s) Found in: Integration Sequence sequence(s)

group Small Group Activity

60 min.

##### Gravitational Potential Energy
Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.

group Small Group Activity

30 min.

##### Charged Sphere
Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

None

Task: Draw a right triangle. Put a circle around the right angle, that is, the angle that is $\frac\pi2$ radians.

• Complete the assignment using your choice of technology. You may write your answers on paper, write them electronically (for instance using xournal), or typeset them (for instance using LaTeX).
• If using software, please export to PDF. If writing by hand, please scan your work using the AIMS scanner if possible. You can also use a scanning app; Gradescope offers advice and suggested apps at this URL. The preferred format is PDF; photos or JPEG scans are less easy to read (and much larger), and should be used only if no alternative is available.)
• Please make sure that your file name includes your own name and the number of the assignment, such as "Tevian2.pdf."

Using Gradescope: We will arrange for you to have a Gradescope account, after which you should receive access instructions directly from them. To submit an assignment:

2. Select the appropriate course, such as "AIMS F21". (There will likely be only one course listed.)
3. Select the assignment called "Sample Assignment"
5. You will then be prompted to associate submitted pages with problem numbers by selecting pages on the right and questions on the left. (In this assignment, there is only one of each.) You may associate multiple problems with the same page if appropriate.
6. When you are finished, click "Submit"
7. After the assignments have been marked, you can log back in to see instructor comments.

• Found in: AIMS Maxwell course(s)

group Small Group Activity

30 min.

##### Stokes' Theorem
Students compute both sides of Stokes' theorem.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

group Small Group Activity

60 min.

##### The Wire
Students compute a vector line integral, then investigate whether this integral is path independent.

• Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

group Small Group Activity

60 min.

##### Multivariable Pictionary
Students draw the 3D graphs of equations using three variables. They make choices for drawing a stack of curves in parallel planes and a curve in a perpendicular plane (e.g. substituting in values for $x$, $y$, or $z$. )

• Found in: Surfaces/Bridge Workshop course(s)

group Small Group Activity

30 min.

##### Grey space capsule
In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

• Found in: Contemporary Challenges course(s)

group Small Group Activity

30 min.

##### Conic Sections
Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.
• Found in: Central Forces course(s)

group Small Group Activity

30 min.

##### Finding $d\boldsymbol{\vec{r}}$
• Found in: Vector Calculus II course(s)

None

##### Centrifuge
A circular cylinder of radius $R$ rotates about the long axis with angular velocity $\omega$. The cylinder contains an ideal gas of atoms of mass $M$ at temperature $T$. Find an expression for the dependence of the concentration $n(r)$ on the radial distance $r$ from the axis, in terms of $n(0)$ on the axis. Take $\mu$ as for an ideal gas.
• Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear

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.

• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop course(s) Found in: Integration Sequence sequence(s)

group Small Group Activity

30 min.

##### The Grid

• Found in: Vector Calculus II course(s)

group Small Group Activity

30 min.

##### The Pretzel

• Found in: Vector Calculus II course(s)