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Activities

Small Group Activity

30 min.

##### Applying the equipartition theorem
Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• Found in: Contemporary Challenges course(s)

Lecture

30 min.

##### Equipartition theorem
This lecture introduces the equipartition theorem.
• Found in: Contemporary Challenges course(s)

Small Group Activity

30 min.

##### Heat capacity of N2
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• Found in: Contemporary Challenges course(s)

Small Group Activity

30 min.

##### Stokes' Theorem
Students compute both sides of Stokes' theorem.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s) Found in: Workshop Presentations 2023 sequence(s)

Small Group Activity

30 min.

##### The Grid
1. Consider the rectangle in the first quadrant of the $xy$-plane as in the figure with thick black lines.
• Label the bottom horizontal edge of the rectangle $y=c$.
• Label the sides of the rectangle $\Delta x$ and $\Delta y$.
• What is the area of the rectangle?
• There are also 2 rectangles whose base is the $x$-axis, the larger of which contains both the smaller and the original rectangle. Express the area of the original rectangle as the difference between the areas of these 2 rectangles.
2. On the grid below, draw any simple, closed, piecewise smooth curve $C$, all of whose segments $C_i$ are parallel either to the $x$-axis or to the $y$-axis. Your curve should not be a rectangle. Pick an origin and label it, and assume that each square is a unit square.
• Compute the area of the region $D$ inside $C$ by counting the number of squares inside $C$.
• Evaluate the line integral $\displaystyle \oint_C y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}$ by noticing that along each segment either $x$ or $y$ is constant, so that the integral is equal to $\sum_{C_i} y\,\Delta x$.

Can you relate this to Problem 1?

• Are your answers to the preceding two calculations the same?
• Would any of your answers change if you replaced $y\,\boldsymbol{\hat{x}}$ by $x\,\boldsymbol{\hat{y}}$ in part (b)?

#### Main ideas

• Understanding different ways of expressing area using integration.
• Concrete example of Area Corollary to Green's/Stokes' Theorem.

We originally used this activity after covering Green's Theorem; we now skip Green's Theorem and do this activity shortly before Stokes' Theorem.

#### Prerequisites

• Familiarity with line integrals.
• Green's Theorem is not a prerequisite!

#### Warmup

• The first problem is a good warmup.

#### Props

• whiteboards and pens
• a planimeter if available

#### Wrapup

• Emphasize the magic -- finding area by walking around the boundary!
• Point out that this works for any closed curve, not just the rectangular regions considered here.
• Demonstrate or describe a planimeter, used for instance to measure the area of a region on a map by tracing the boundary.

### Details

#### In the Classroom

• Make sure students use a consistent orientation on their path.
• Make sure students explicitly include all segments of their path, including those which obviously yield zero.
• Students in a given group should all use the same curve.
• Students should be discouraged from drawing a curve whose longest side is along a coordinate axis.
• Students may need to be reminded that $\oint$ implies the counterclockwise orientation. But it doesn't matter what orientation students use so long as they are consistent!
• A geometric argument that the orientation should be reversed when interchanging $x$ and $y$ is to rotate the $xy$-plane about the line $y=x$. (This explains the minus sign in Green's Theorem.)
• Students may not have seen line integrals of this form (see below).
• Students do very well on this lab, particularly after working in groups for several weeks. Resist the urge to intervene.
• Make sure everyone sees the reason $y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}$ is zero on vertical pieces.
• The issue of the negative will come up. Suggest students make a quick sketch of the vector field.
• It is well worthwhile to do an example with a circle together as a class. The line integral should pose no trouble for them and the area of a circle is something they believe.
• Emphasize the connection between the boundary and the interior. This is a concrete display of this relationship.

#### Subsidiary ideas

• Orientation of closed paths.
• Line integrals of the form $\int P\,dx+Q\,dy$.

We do not discuss such integrals in class! Integrals of this form almost always arise in applications as $\int\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}$.

#### Homework

Determine the area of a triangle or an ellipse by integrating along the boundary.

#### Essay questions

Describe times in your life when you needed to know area (or imagine such a time). Maybe buying carpet or painting a room. What is the first step in computing area? How does this lab truly differ, if at all?

#### Enrichment

• Write down Green's Theorem.
• Go to 3 dimensions --- bend the curve out of the plane and stretch the region like a butterfly net or rubber sheet. This is the setting for Stokes' Theorem!
• Found in: Vector Calculus II course(s)