Path integrals and special vector fields #

Background #

The goal of this lab is to explore a number of properties of vector fields suggested by our in-class work. Here are the questions we will examine:

• Does the path integral of a vector field over a fixed curve depend on how we parameterize that curve?
• Does the path integral of a vector field from point \(A\) to point \(B\) depend on which curve from \(A\) to \(B\) we choose, or is it completely determined by the endpoints?
• When is the path integral of a vector field around a closed curve equal to zero?

In this lab, we will collect experimental data by computing many path integrals over a variety of curves and vector fields and form conjectures as to what might be true in general. When we reconvene in class, we will analyze our results.

Experimental setup #

Our computational work will be done using a Mathematica notebook. It defines a variety of vector fields and curves to help you answer the three questions above. You can plot both the vector fields and the curves using commands in the notebook as well as compute path integrals without breaking a sweat. In each section you can tabulate your results in a provided table.

Independence of Parametrization #

The first issue on our agenda is the question of whether path integrals depend on how we choose to parameterize the underlying curves. More precisely:

For example,

\begin{align*} \vec{r}_1(t) & = t \; \vec{i} + (1 - t^2) \; \vec{j} & -1 &\leq t \leq 1\\ \vec{r}_2(t) & = (t^2 - 1) \;\vec{i} + t^2(2-t^2) \; \vec{j} & 0 & \leq t \leq \sqrt{2}\\ \vec{r}_3(t) & = -\cos t \; \vec{i} + \sin^2 t \; \vec{j} & 0 & \leq t \leq \pi \\ \end{align*}

are three different parameterizations of the same curve which begins at \((-1,0)\) and ends at \((1, 0)\). Graph all three parametrizations using Mathematica and verify that they indeed represent the same curve.

Independence of Path #

The next question we need to answer is whether the value of the line integral depends on the actual path, or only on its beginning and ending points. More precisely,

For our experiment, we need a variety of curves that all begin and end at the same points. If we take \(-1 \leq t \leq 1\), then the following four curves all begin at \((-1, 0)\) and all end at \((1,0)\). \begin{align*} \vec{s}_1(t) & = t \vec{i} + (1-t^2)\vec{j} \\ \vec{s}_2(t) & = t \vec{i} + .7(t -1)(t + 1)(t - .7) \vec{j} \\ \vec{s}_3(t) & =(t^2 + t - 1) \vec{i} + \sin(2\pi t) \vec{j} \\ \vec{s}_4(t) & = (t + \sin^2(2\pi t)) \vec{i} + (1- e^{1 - t^2}) \vec{j} \\ \end{align*}

Begin by plotting all four of the curves above, verify that they all begin and end at the points we claim, and that they are indeed different from each other.

Circulation-free vector fields #

A vector field \(\vec{F}\) is said to be circulation free if the path integral of \(\vec{F}\) around every closed curve equals zero. We are interested in the following question:

Consider the following four closed curves, all defined on the interval \(0 \leq t \leq \pi\).

\begin{align*} \vec{s}_5(t) & = (\cos t + 0.5 \cos 2t) \vec{i} + (0.5\sin(t) - 1) + \tfrac{1}{6} \cos(3t))\vec{j} \\ \vec{s}_6(t) & = (0.8 + \cos t)\vec{i} + (0.5\sin(t) - 0.25)\vec{j} \\ \vec{s}_7(t) & = (\cos(2t)\sin(\tfrac{1}{2}t))\vec{i} + (1 + 0.5\sin(2t)\sin(\tfrac{1}{2}t))\vec{j}\\ \vec{s}_8(t) & = \big(t(t - \tfrac{2\pi}{10})\big) \vec{i} + \big(1 + \tfrac{t(t - 2\pi)(t - 1)}{30}\big) \vec{j} \\ \end{align*}

First, check that each of the curves is truly closed by plotting and then continue to the following exercise.

Irrotational vector fields #

Suppose that we have written a vector field as \(\vec{F}(x,y) = P(x,y)\vec{i} + Q(x,y)\vec{j}\) where \(P(x,y)\) and \(Q(x,y)\) are functions of \(x\) and \(y\). The curl of \(\vec{F}\) is defined to be the quantity \[ \text{curl} \; \vec{F}(x,y) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}.\]

At first glance, the curl may seem like a fairly strange quantity to study, but I am hoping that you will indulge me for the time being. A vector field is called irrotational if its curl is always equal to zero.

Note that the way we have defined the adjectives path-independent and circulation-free, we are actually unable to check whether a vector field is either. We simply cannot verify that this property holds for all revelant paths. But unlike other adjectives in this lab, we can actually find out whether a vector field is irrotational or not: we just have to compute two derivatives!

The final exercise is the crux of this lab. Spend some time thinking about it and then write up your answer as part of the homework.

This lab is based on original work by William H. Barker.