Activities
- to perform a magnetic vector potential calculation using the superposition principle;
- to decide which form of the superposition principle to use, depending on the dimensions of the current density;
- how to find current from total charge \(Q\), period \(T\), and the geometry of the problem, radius \(R\);
- to write the distance formula \(\vec{r}-\vec{r'}\) in both the numerator and denominator of the superposition principle in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.
Students work in small groups to interpret level curves representing different concentrations of lead.
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In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.
This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..
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 use \(d\boldsymbol{\vec{A} }= d\boldsymbol{\vec{r}}_1 \times d\boldsymbol{\vec{r}}_2\) and \(d\tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\) to find differential surface and volume elements for cylinders and spheres.
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- Vectors and their magnitudes are geometric quantities, independent of coordinates and choice of basis
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- Outer products yield projection operators
- Projection operators are idempotes (they square to themselves)
- A complete set of outer products of an orthonormal basis is the identity (a completeness relation)
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
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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 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.
A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]
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.
- Recall the relationship between the sign of the dot product and the orientation of the vectors.
- Use graphical methods to estimate the value of a vector line integral.
- Lay the groundwork for thinking about conservative and non-conservative vector fields.
- Practice evaluating line integrals;
- Practice choosing appropriate coordinates and basis vectors;
- Introduction to the geometry behind conservative vector fields.
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 learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates
In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
- A component of the curl of a vector field (at a point) is the circulation per unit area around an infinitesimal loop.
- How to predict the sign and relative magnitude of the curl from graphs of a vector field.
- (Optional) How to calculate the curl of a vector field using computer algebra.
- Divergence of a vector field (at a point) is the flux per unit volume through an infinitesimal box.
- How to predict the sign and relative magnitude of the divergence from graphs of a vector field.
- (Optional) How to calculate the divergence of a vector field with computer algebra.
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.
- How to form a state as a column vector in matrix representation.
- How to do probability calculations on all three representations used for quantum systems in PH426.
- How to find probabilities for and the resultant state after measuring degenerate eigenvalues.
Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
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 vector line integrals and explore their properties.
Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.
Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.
In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.