assignment_ind Small White Board Question

10 min.

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

group Small Group Activity

10 min.

##### Surfaces Intro

An introduction to the use of the Raising Calculus surfaces.

Students have to match their surface with the appropriate contour map.

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

None

##### The Cube
Find the angle between the diagonal of a cube (connecting opposite corners) and the diagonal of one of its faces (connecting opposite corners of one square face).
• Found in: Vector Calculus I, 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)

face Lecture

5 min.

##### Multivariable Differentials
This mini-lecture demonstrates the relationship between $df$ on the tangent plane to its “components“ in coordinate directions, leading to the multivariable chain rule.

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

None

##### Total Current, Square Cross-Section
1. Current $I$ flows down a wire with square cross-section. The length of the square side is $L$. If the current is uniformly distributed over the entire area, find the current density .
2. If the current is uniformly distributed over the outer surface only, find the current density .
• Found in: Integration Sequence sequence(s) Found in: Static Fields, AIMS Maxwell course(s)

group Small Group Activity

30 min.

##### Chain Rule Measurement
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the $r$-direction and derivatives in $x$- and $y$-directions using the chain rule.
• Found in: Vector Calculus I course(s)

None

##### Current from a Spinning Cylinder
A solid cylinder with radius $R$ and height $H$ has its base on the $x,y$-plane and is symmetric around the $z$-axis. There is a fixed volume charge density on the cylinder $\rho=\alpha z$. If the cylinder is spinning with period $T$:
1. Find the volume current density.
2. Find the total current.

computer Mathematica Activity

30 min.

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop course(s) Found in: Gradient Sequence sequence(s)

computer Computer Simulation

30 min.

##### Blackbody PhET
Students use a PhET to explore properties of the Planck distribution.

• Found in: Contemporary Challenges course(s)

group Small Group Activity

5 min.

##### Acting Out Flux
Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

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

group Small Group Activity

5 min.

##### Finding a Chain Rule
Students use chain rule diagrams to construct a multivariable chain rule in terms of differentials.
• Found in: Surfaces/Bridge Workshop course(s)

None

##### Total Current, Circular Cross Section

A current $I$ flows down a cylindrical wire of radius $R$.

1. If it is uniformly distributed over the surface, give a formula for the surface current density $\vec K$.
2. If it is distributed in such a way that the volume current density, $|\vec J|$, is inversely proportional to the distance from the axis, give a formula for $\vec J$.

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

group Small Group Activity

30 min.

##### Ideal Gas Model
Students consider whether the thermo surfaces reflect the properties of an ideal gas.

group Small Group Activity

30 min.

##### Directional Derivatives
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

• Found in: Vector Calculus I course(s) Found in: Gradient Sequence sequence(s)

computer Computer Simulation

30 min.

##### Visualizing Flux through a Cube
Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.
• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop course(s)

None

##### Divergence

Shown above is a two-dimensional vector field.

Determine whether the divergence at point A and at point C is positive, negative, or zero.

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

group Small Group Activity

30 min.

##### Number of Paths
Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.

group Small Group Activity

5 min.

##### The Resistors
This small group activity is designed to provide practice with the multivariable chain rule. Students determine a particular rate of change using given information involving other rates of change. The discussion emphasizes the equivalence of a variety of approaches, including the use of differentials. Good “review” problem; can also be used as a homework problem.
• Found in: Vector Calculus I, Surfaces/Bridge Workshop course(s)

None

##### 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}

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