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Calculus & Mathematica: 254 Course Description

Vector Calculus

VC.01 Vectors Point the Way

Mathematics

Vectors: How they are plotted, how they are moved, how they are added and how they are multiplied by numbers. Tangent vectors, velocity vectors and tangent lines. Dot product of two vectors and the component (push) of one vector in the direction of another.

Science and Math Experience

Bouncing light beams off curves in two dimensions. Experiments dealing with reflecting properties of parabolas and ellipses. Pursuit models. Parametric formulas for a line. Laser zapping along the tangent line. Velocity and acceleration vectors. Analyzing an object's motion and speed by looking at tangential components of its acceleration. Normal and tangential components of acceleration. Plotting the motion of the Earth-Mars-Sun system with Earth at the origin. Stealth technology.

VC.02 Perpendicularity

Mathematics

Cross product and dot products. Planes in three dimensions. Normal vectors for surfaces in three dimensions.

Science and Math Experience

Flatness and plotting. Using the main unit normal and binormal as moving frames to plot tubes, horns, ribbons and corrugations centered on a given curve. Experiments with linearizations (tangent plane approximations). Bouncing light beams off surfaces in three dimensions. Kissing circles and curvature in two and three dimensions. Boring holes with a robotic router.

VC.03 Gradient

Mathematics

The gradient and the chain rule. Level curves, level surfaces and the gradient as normal vector. The gradient as a vector pointing in the direction of greatest initial increase. Linearizations and the chain rule. Total differential. LaGrange multipliers.

Science and Math Experience

Using the gradient for maximization and minimization. Ascent and descent paths through the gradient field with application to the idea behind Mathematica's FindMinimum instruction. Data Fit: Vibrating string and one-dimensional heat. Programming heat-seeking missiles. Cobb-Douglas manufacturing model. Experience with Beale's valley function, Rosenbrock's banana function. Optimization problems from metallurgy. Duffin's barge problem.

VC.04 2D Vector Fields and Their Trajectories

Mathematics

Vector fields as fluid flow. Trajectories in vector fields as the path a cork floats on. Solutions of differential equations y'[x] = f[x,y[x]] as trajectories in a vector field.

Science and Math Experience

Tangential and normal components of vector fields on given curves. Visual experiments dealing with the net flow of vector fields across given curves. Visual experiments dealing with the net flow of vector fields along given curves. The 2D electric field. Dipoles in 2D. The gradient field. Experiments with how gradient fields look near maximizers and minimizers. Where trajectories in the gradient field want to go. Where trajectories in the negative gradient field want to go. Looking for spigots and drains by following the path of a cork. Logistic harvesting model.

VC.05 Flow Measurements

Mathematics

Path integrals (line integrals) as measurements of the net flow of a given vector field across a given curve. Path integrals (line integrals) as measurements of the net flow of a given vector field along a given curve. Path independence and gradient fields. Recognition of gradient fields.

Science and Math Experience

Sources and sinks of 2D vector fields. Sources and sinks at singularities in vector fields. Work as flow in a force field. Which way to go to make the a force field do most of the work. Force fields and their trajectories. Models for water flow. Clockwise versus counterclockwise flow.

VC.06 Sources, Sinks, Swirls, and Singularities

Mathematics

Divergence and rotation of a 2D vector field. Sources as points at which the divergence is positive; sinks as points at which the divergence is negative. Using the Gauss-Green formula to measure the flow of a 2D vector field across a closed curve by means of a 2D integral. Using the Gauss-Green formula to measure the flow of a 2D vector field along a closed curve by means of a 2D integral. Singularity sources, sinks and swirls.

Science and Math Experience

Why it is that if all points inside a given closed curve C are sources of a given 2D vector field, then the net flow of the vector field across C must be from inside to outside. Encapsulating singularities with small circles centered on the singularity. Flow measurements in the presence or absence of singularities. 2D electric fields. Dipole fields. Gauss's law for calculating the flux of combined 2D electric fields. Parallel flow. The Laplacian as the divergence of the gradient field. Why harmonic functions cannot have local maxima or minima. Steady state heat and the Laplace's equation in two dimensions. Spin fields.

VC.07 Transforming 2D Integrals

Mathematics

uv paper and xy paper when u = u[x,y] and v = v[x,y]. The uv grid on xy paper. Linearizing the uv grid on xy paper. The Jaocbian as the area conversion factor for converting xy paper area measurements into uv paper area measurements. Transforming 2D integrals: How it is done and why it is done.

Science and Math Experience

Semi-log and log-log paper. Flow measurements for 2D vector fields. If the boundary of a region can be plotted with Mathematica, then the plotting instructions usually carry enough information to make it possible to measure the area of the region. Why crazy things are likely to happen when the area conversion factor is 0. Analyzing a transformation by plotting its area conversion factor and by plotting the gradient field of its area conversion factor. Experiments relating linear equations and area measurements in two dimensions. Experiments with eigenvectors as stretching directions and eigenvalues as stretching factors in linear transformations in two dimensions.

VC.08 Transforming 3D Integrals

Mathematics

The 3D integral _R f[x,y,z] dx dy dz via slicing and accumulating. _R dx dy dz as a volume measurement. Average value of a function on a region. The Jacobian as a local volume conversion factor in 3D. Transforming 3D integrals. Mass and density.

Science and Math Experience

If the whole skin of a solid region can be plotted with Mathematica, then the plotting instructions usually carry enough information to make it possible to measure the volume of the solid region. Cylinders, spheres and tubes: Plotting them and integrating on them. Integrating on solids bounded by sets of surfaces. Switching the order of integration. Tubes, horns and squashed doughnuts. Drilling and slicing spheres. Experiments relating linear equations to volume measurements. Centroids and centers of mass. Bidding on rocket nose cones.

VC.09 Spherical Coordinates

Mathematics

Meaning of each of the spherical parameters. Plotting and measuring volumes of spheres, ellipsoids, cones and measuring volumes of each. Integration with spherical coordinates.

Science and Math Experience

Earth-Moon animations. Snail shells. Star Wars window of vulnerability plots adapted from NASA work. Using spherical coordinates to design and paint flowers. Centering and aligning the general 3D ellipsoid. Ice cream cone and tops. Passing a plane between two disjoint solid disks. Measuring the volume inside 4D spheres. Experiments with eigenvectors as stretching directions and eigenvalues as stretching factors in linear transformations in three dimensions.

VC.10 3D Surface Measurements

Mathematics

Measuring area on surfaces. Surface integrals for measuring flow of 3D vector fields across surfaces. Sources, sinks and Gauss's formula in 3D.

Science and Math Experience

Measuring flow across surfaces: Gauss's formula versus calculation by a surface integral. Substitute surfaces to avoid calculational nightmares. Encapsulating singularities with small spheres centered on the singularity. 3D electric fields and Gauss's law for calculating the flux of combined 3D electric fields. The Laplacian as the divergence of the 3D gradient field. Sources and sinks in the 3D gradient field and their relation to max-min. Why harmonic functions of three variables cannot have local maxima or minima. Steady state heat in a solid. Morphing and Moebius strips.

VC.11 3D Flow Along Measurements

Mathematics

Measuring flow along a 3D curve with a path integral. The curl of a 3D vector field. Orientation and Stokes's formula. Stokes's formula as an outgrowth of the 2D Gauss-Green formula.

Science and Math Experience

Fingering a 3D vector field. The curl as the axis of the greatest counterclockwise swirl. Paddle wheels. Parallel flow and irrotational flow. Ideal fluid flow. Work done by 3D force fields. Recognition of 3D gradient fields. Path independence.

Significant Changes From the Traditional Course

• The main change is the visual, conceptual approach. Students learn the basic ideas visually and then they learn the about the integrals that make the associated measurements. The result: Instead of studying the symbols in a formula, the students learn they meaning of a given measurement formula and when to apply it.
• One obvious change is the students' ability to plot what ever they want. In the traditional course, visualization of two dimensional regions and three dimensional curves and surfaces has always been a problem. The traditional response has been to look only at surfaces that have circular, elliptical, hyperbolic or parabolic cross sections. There is no such restriction in this course.
• Instead of delivering most surfaces implicitly or explicitly, this course delivers most surfaces parametrically.
• Vectors are used right at the beginning as strong motivators in experiments involving laser burns, pursuit models and planet plotting-first with the sun at the origin and second with Earth at the origin.
• The reflecting properties of parabolic and elliptical reflectors are discovered experimentally by the students and explained through student calculations.
• Students learn about the principal unit normal and the unit binormals and they retain the idea because they use these two normal vectors to plot tubes, horns and ribbons centered on curves. The value of this plotting experience as a strong driver the acquisition and retention of the idea of moving frames cannot be underestimated.
• The idea of curvature is developed cleanly with vector ideas so that the two dimensional version can be immediately transported to three dimensions. For the first time in a calculus course, students plot actual curves and osculating circles in two and three dimensions.
• The meaning of velocity vectors and acceleration vectors, the tangential components of the acceleration vectors and the normal components of acceleration vectors are set-up with student produced plots and write-ups. Instead of looking at the dot product and cross product as isolated topics, the dot product and cross product are studied under the unifying theme of perpendicularity.
• Students learn about ascent and descent paths by working with plots of the gradient field. The idea is reinforced experiments like programming a heat seeking missile on an ascent path through the gradient field of the temperature function.
• Students get experience in optimization by working with Beale's valley function, Rosenbrock's banana function, optimization problems from metallurgy and least squares data fit in more than one dimension.
• Local maximizers and local minimizers are de-emphasized in favor emphasis of global maximizers and global minimizers.
• Vector fields are visually introduced as flow models. Students drop floating corks into plots of vector fields with the goal of spotting sources and sinks. Students plot normal and tangential components of vector fields on curves with the goal of coming up with their own visual estimations of the net flow of a given vector field across a given curve (flux) and the net flow of a given vector field along a given curve (circulation or rotation). After the ideas are established in the students' minds visually, the associated measurements coming from path integrals (line integrals) and the Gauss-Green formula are introduced.
• Students learn to use the divergence and curl of a vector field in a conceptual way as opposed to the traditional algebraic way. They are able to work with a formula and the meaning of the formula interchangeably. For instance, upon learning that the divergence of a given vector field Field[x,y] = {m[x,y], n[x,y]} is positive at all points within a given curve C, the student will say that the all the points inside C are sources of new fluid. They will go on to say that the net flow of this vector field across C is from inside to outside and consequently the net flow-across-C measurement _C -n[x,y] dx + m[x,y] dy is positive.
• The Laplacian is introduced as a measurement for detecting sources and sinks in the gradient field. Students explain why functions that measure steady state heat must have be vanishing Laplacians. Similarly students explain, in two and three dimensions, why a function whose Laplacian vanishes cannot have maximum or minimum values.
• Students get experience with electric fields in two and three dimensions by plotting them and making measurements on them. Students explain the reasoning behind Gauss's law for the electric flux. Students have reported that this experience has been very helpful to them in their physics courses.
• The two-dimensional Jacobian is introduced a local area conversion factor; the three-dimensional Jacobian is introduced as a local volume conversion factor. Students study the action of transformations by plotting Jacobians and their gradient fields and learn why a Jacobian that vanishes at a point can cause big problems.
• Students learn how to react to a given 2D or 3D integral and to come up with a custom transformation of a nasty 2D or 3D integral into an easily calculated integral. Traditional courses usually limit themselves to polar, cylindrical or spherical transformations of integrals. There is no such restriction here. For traditional students, integrating on an ellipsoid is a big deal; for students in this course, integrating on an ellipsoid is routine. Reason: Students know that if they can plot a surface of a solid parametrically, then usually the plotting instructions set up the transformation they want to use to integrate on or within the solid.