This follows Equipartition theorem

If the microscopic world was classical, predict \(U_{\text{classical}}(T)\) for the following “toy molecules” in the gas phase.

(a) (b) (c) (d)

• Each ball is a point mass \(m\) with no moment of inertia.
• The zig-zag lines are springs which are freely jointed at the balls.
• Vibrational motion of the springs is very small (\(\ll\) the length of the spring).
• The springs can extend and compress, but cannot twist or flex.
• The straight lines are rigid rods.

This lecture introduces the equipartition theorem.

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.”

Students compute both sides of Stokes' theorem.

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!