Activities
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.
Students compute a vector line integral, then investigate whether this integral is path independent.
For each of the following vector fields, find a potential function if one exists, or argue that none exists.
- \(\boldsymbol{\vec{F}} = (3x^2 + \tan y)\,\boldsymbol{\hat{x}} + (3y^2 + x\sec^2 y) \,\boldsymbol{\hat{y}}\)
- \(\boldsymbol{\vec{G}} = y\,\boldsymbol{\hat{x}} - x\,\boldsymbol{\hat{y}}\)
- \(\boldsymbol{\vec{H}} = (2xy + y^2 \sin z) \,\boldsymbol{\hat{x}} + (x^2 + z + 2xy\sin z) \,\boldsymbol{\hat{y}} + (y + z + xy^2 \cos z) \,\boldsymbol{\hat{z}}\)
- \(\boldsymbol{\vec{K}} = yz \,\boldsymbol{\hat{x}} + xz \,\boldsymbol{\hat{y}}\)
Main ideas
- Finding potential functions.
Students love this activity. Some groups will finish in 10 minutes or less; few will require as much as 30 minutes. ^{*}
Prerequisites
- Fundamental Theorem for line integrals
- The Murder Mystery Method
Warmup
none
Props
- whiteboards and pens
Wrapup
- Revisit integrating conservative vector fields along various paths, including reversing the orientation and integrating around closed paths.
Details
In the Classroom
- We recommend having the students work in groups of 2 on this activity, and not having them turn anything in.
- Most students will treat the last example as 2-dimensional, giving the answer \(xyz\). Ask these students to check their work by taking the gradient; most will include a \(\boldsymbol{\hat{z}}\) term. Let them think this through. The correct answer of course depends on whether one assumes that \(z\) is constant; we have deliberately left this ambiguous.
- It is good and proper that students want to add together multivariable terms. Keep returning to the gradient, something they know well. It is better to discover the guidelines themselves.
Subsidiary ideas
- 3-d vector fields do not necessarily have a \(\boldsymbol{\hat{z}}\)-component!
Homework
A challenging question to ponder is why a surface fails to exist for nonconservative fields. Using an example such as \(y\,\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}}\), prompt students to plot the field and examine its magnitude at various locations. Suggest piecing together level sets. There is serious geometry lurking that entails smoothness. Wrestling with this is healthy.
Essay questions
Write 3-5 sentences describing the connection between derivatives and integrals in the single-variable case. In other words, what is the one-dimensional version of MMM? Emphasize that much of vector calculus is generalizing concepts from single-variable theory.
Enrichment
The derivative check for conservative vector fields can be described using the same type of diagrams as used in the Murder Mystery Method; this is just moving down the diagram (via differentiation) from the row containing the components of the vector field, rather than moving up (via integration). We believe this should not be mentioned until after this lab.
When done in 3-d, this makes a nice introduction to curl --- which however is not needed until one is ready to do Stokes' Theorem. We would therefore recommend delaying this entire discussion, including the 2-d case, until then.
- Work out the Murder Mystery Method using polar basis vectors, by reversing the process of taking the gradient in this basis.
- Revisit the example in the Ampère's Law lab, using the Fundamental Theorem to explain the results. This can be done without reference to a basis, but it is worth computing \(\boldsymbol{\vec\nabla}\phi\) in a polar basis.
Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
Students work in small groups to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
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.
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
Gauss's Law: \[ \oint \vec{E}\cdot \hat{n}\, dA = {1\over\epsilon_0}\, Q_{\hbox{enc}} \]
Ampère's Law:
\[ \oint \vec{B}\cdot d\vec{r} = \mu_0 \, I_{\hbox{enc}} \]
Potentials: \begin{eqnarray*} \vec{E}&=&-\vec{\nabla} V\\ \vec{B}&=&\vec{\nabla}\times\vec{A} \end{eqnarray*}
Maxwell's Equations: \begin{eqnarray*} \vec{\nabla}\cdot\vec{E} &=& \frac{\rho}{\epsilon_0}\\ \vec{\nabla}\cdot\vec{B} &=& 0\\ \vec{\nabla}\times\vec{E} &=& 0\\ \vec{\nabla}\times\vec{B} &=& {\mu_0}\, \vec{J} \end{eqnarray*}
Superposition Laws: \begin{eqnarray*} V(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{E}(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')(\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ \vec{A}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{B}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\times (\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ V(B)-V(A)&=&-\int_A^B \vec{E}\cdot d\vec{r} \end{eqnarray*}
Position Vectors \begin{align*} \vec{r} &= x \hat{x} + y\hat{y} + z\hat{z}\\ &= s \hat{s} + z\hat{z}\\ &= r\hat{r} \end{align*}The distance between two position vectors
- In cylindrical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{s^2+s^{\prime\, 2}-2s\, s^{\prime}\cos(\phi- \phi^{\prime}) +(z-z^{\prime})^2}\]
- In spherical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{r^2+r^{\prime\, 2}-2r\, r^{\prime}\left[ \sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime}) +\cos\theta\cos\theta^{\prime}\right]}\]
Rectangular Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial x}\,\hat{x} + \frac{\partial f}{\partial y}\,\hat{y} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x} + \left(\frac{\partial F_x}{\partial z} -\frac{\partial F_z}{\partial x}\right)\hat{y} + \left(\frac{\partial F_y}{\partial x} -\frac{\partial F_x}{\partial y}\right)\hat{z} \end{eqnarray*}
Cylindrical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial s}\,\hat{s} + \frac{1}{s}\frac{\partial f}{\partial \phi}\,\hat{\phi} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{s}\frac{\partial}{\partial s}\left({s}F_{s}\right) + \frac{1}{s}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left( \frac{1}{s}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right) \hat{s} + \left(\frac{\partial F_s}{\partial z}-\frac{\partial F_z}{\partial s}\right) \hat{\phi} + \frac{1}{s} \left( \frac{\partial}{\partial s}\left({s}F_{\phi}\right) - \frac{\partial F_s}{\partial \phi} \right) \hat{z} \end{eqnarray*}
Spherical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial r}\,\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\,\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\,\hat{\phi} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{r^2}\frac{\partial}{\partial r}\left({r^2}F_{r}\right) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left({\sin\theta}F_{\theta}\right) + \frac{1}{r\sin\theta}\frac{\partial F_\phi}{\partial \phi} \\ \vec{\nabla}\times\vec{F} &=& \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left({\sin\theta}F_{\phi}\right) - \frac{\partial F_\theta}{\partial \phi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}\left({r}F_{\phi}\right) \right) \hat{\theta} \\ && \quad + \frac{1}{r} \left( \frac{\partial}{\partial r}\left({r}F_{\theta}\right) - \frac{\partial F_r}{\partial \theta} \right) \hat{\phi} \end{eqnarray*}
Lorentz Force Law:\[\vec{F}=q_{\hbox{test}}\left(\vec{E}+\vec{v}\times\vec{B}\right)\]
Step and Delta Functions: \begin{eqnarray*} \frac{d}{dx} \theta(x-a)&=&\delta(x-a)\\ \int_{-\infty}^{\infty} f(x)\delta(x-a)\, dx&=&f(a) \end{eqnarray*}
Vector Calculus Theorems: \begin{eqnarray*} \oint \vec{F} \cdot d\vec{A} &=& \int \vec{\nabla} \cdot \vec{F} d\tau\\ \oint \vec{F} \cdot d\vec{\ell} &=& \int (\vec{\nabla} \times \vec{F}) \cdot d\vec{A}\\ \end{eqnarray*}
Total Charge and Current: \begin{eqnarray*} Q &=& \int \rho (\vec{r}') d\tau'\\ I &=& \int \vec{J} (\vec{r}') \cdot d\vec{A'}\\ \end{eqnarray*}
Sketch each of the vector fields below.
- \(\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
- \(\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}\)
- \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.