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Math 242

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2D and 3D Measurements and Expansions
Authors:  Bill Davis, Horacio Porta and Jerry Uhl  ©1999
Producer:  Bruce Carpenter
Publisher:          Distributor:  

VC.01 Vectors Point the Way

Mathematics.

Vectors. Adding and subtracting vectors.   Tangent vectors, velocity vectors, and tangent lines.Length of a vector, dot product, and distance between two points. Perpendicular vectors. The push of one vector in the direction of another, and the formula
       ,
where is the angle between and .

Science and math experience.

Velocity and acceleration. Bouncing light rays off curves. Pursuit models. Laser gunnery.  Planetary motion.  Parabolic reflectors, spherical reflec tors, and elliptical reflectors. Stealth technology.

VC.02 Perpendicularity

Mathematics.

The cross product of two 3D vector s. Lines and planes in 3D.  Normal vectors for curved surfaces in 3D. Main unit normals, binormals.

Science and math experience.

Plotting tubes and corrugations. Using the normal vector to bounce light beams off surfaces. Kissing circles and curvature. Measurements with the cross product. Breakin g acceleration vectors into normal and tangential components. Programming routers to cut specified pieces out of sheet steel. Plotting on planes. Serious 3D plots: Tube s and ribbons.

VC.03 The Gradient

Mathematics.

The gradient and the chain rule. Level curves, level surfaces and the gradient as normal vector. The gradient points in the direction of greatest initial increase.Using linearizations to help to explain the chain rule. How to use the gradient for maximization and minimization. The total differential.

Science and math experience.

Estimating whether a given function has a global maximizer, minimizer or both. Heat seeking missiles. Closest points, gradients and Lagrange's method. The Cobb-Douglas manufacturing model.  Data Fit in two variables.

VC.04 2D Vector Fields and Their Trajectories
        

Mathematics.

Flow of vector fields along curves.  Flow of vector fields across curves.
Differential equations and and their associated vector fields.

Science and math experience.

Gradient fields try to flow toward maximizers. . Looking for sinks (drains). Electric Fields.  Normal components and flow across curves. Tangential components and flow along curves. Logistic harvesting. Plotting level curves. Gradient field versus Hamiltonian field.

VC.05 Flow Measurements by Integral

Mathematics.

Measuring flow across a curve with the integral   
       .
Measuring flow along a curve with the integral
  ;      .
Measurements made with path integrals  
       .
Directed curves; path integrals, path independence and gradient fields.
Recognizing gradien t fields:  the gradient test.

Science and math experience.

Path integrals: backwards and forwards. Water. Sources and sinks. Force fields.  Work and how the physicists measure it.

VC.06 Sources,  Sinks,  Swirls and Singularities
         

Mathematics.

Gauss-Green formula. Using a 2D integral to measure flow across closed curves.
Using a 2D integral to measure flow along closed curves.Using the divergence of a vec tor field to identify sources and sinks.  Flow across a closed curve and flow along a closed curve.  Measurements in the presence of singularities.< br>

Science and math experience.

2D electric fields, dipole fields, and Gauss's law in physics.
       
when
        
        
 &nbs p;The Laplacian and steady-state heat.
Maximum and minimum principle for functions satisfying Laplace's equation
           
  
   Rotation and parallel flow.

VC.07 Transforming 2D Integral

Mathematics.

Going between -paper and -paper. Transforming 2D integrals: How you do it and why you do it. Linearizing the grids. Derivation of area conversion factor (Jacobian) via linearization.

Science and math experience.

How the plotting instructions reveal how to transform wicked 2D integrals into easy 2D integrals. Ribbons. More on flow measurements.  Semi-log paper and log- log paper. What can happen at points at which the the area conversion factor (Jacobian) is zero. What information the sign of the area conversion factor (Jacobian) reve als. Streamlines for flow out of an open pipe. Streamlines for airfoils.

VC.08 Transforming 3D Integrals

Mathematics.

3D integrals. Transforming wicked 3D integrals into easy 3D integrals. Volume measurements through transforming 3D integrals. Average value of a function.

Science and math experience.

How the plotting instructions reveal how to transform wicked 3D integrals into easy 3D integrals. Cylindrical coordinates. Centroids, and centers of mass. Cylinders, sp heres, and tubes:  Plotting them and integrating on them.  Switching the order of integration.  Drilling and slicing spheres. The box prod uct for measuring the volume of 3D parallelepipeds.

VC.09 Spherical Coordinates

Mathematics.

Spherical coordinates. Using spherical coordinates in 3D integration.

Science and math experience.

Using spherical coordinates to plot parts of spheres. Using spherical coordinates to plot cones and other surfaces. Earth-moon plots. Estimating the kill range of mobil e lazer cones.  Inserting planes between disjoint spheres. Spherical coordinate art.  Measurements in four and five dimensions.

VC.10 3D Surface Measurements

Mathematics.

Divergence and Gauss's formula in 3D. Using the 3D divergence to identify sourses and sinks in 3D vector fields. Surface integrals. Using surface integrals to measure f low across 3D surfaces.

Science and math experience.

3D electric fields and Coulomb's law. Gauss's 3D formula versus flow calculation via surface integrals.  Using Gauss's formula to avoid a calculational nightm are: Calculating flow across an oddball surface via calculating the flow across a substitute surface. Using Gauss's formula to take advantage of singularities: Calculat ing flow across the skin of a solid region via calculating the flow across a substitute sphere. Flux of the electric field and Gauss's electric law in 3D. The 3D Laplac ian and steady-state heat.
Ma ximum and minimum principle for functions satisfying Laplace's equation
           .
  

VC.11 3D Flow Along
         

Mathematics.

Measuring flow along a 3D curve via path integrals.The curl of a 3D vector field and what it measures. Orientation of surfaces.  Stokes's  formula i n 3D. Path dependence and independence. The gradient test in 3D.

Science and math experience.

Force fields and the relation between flow-along measurements and work. Using 3D path integrals to construct functions with a given gradient.

3.Approximation

3.01 Splines

Mathematics.

Remarkable plots explained by order of contact.  Splining for smoothness at the knots.

Science and math experience.

Experiments geared at discovering that the smoother the transition from one curve to another at a knot, the better both curves approximate each other near the knot.&nbs p; Splining functions and polynomials.  Splines in road design.  Landing an airplane.  The natural cubic spline.  Order of contact for derivatives and integrals.

3.02 Expansions in Powers of x

Mathematics.

The expansion of a function in pow ers of as a file of polynomials&nb sp; with higher and higher orders of contact with at . &nbs p;The expansions every literate calculus person knows:
       , , , and .
Converting known expansio ns to others via change of variable.
Expansions for approximations.

Science and math experience.

Experiments geared toward discovering that using more and more of the  expansion results in better and better approximation.  Halley's way of & nbsp;calculating accurate decimals of .  Expansions by substitution.  Expansions by differentiation.  Expansions by integration.  Recognition of  expansi ons.  Expansions that satisfy a priori error bounds.

3.03 Using Expansions

Mathematics.

The expansion of a function in pow ers of as a file of  pol ynomials with higher and higher orders of contact with at . &nbs p;Netwon's method.  Multiplying and dividing expansions.  Using expansions to help calculate limits at a point.  Expansions and the comple x exponential function.  Using expansions to help to get precise estimates of some integrals.

Science and math experience.

Centering expansions for good approximation.  Newton's method for root finding.  Successes and failures of Newton's method.  Using the com plex exponential to generate trigonometric identities.  Comparing reflecting properties of spherical mirrors and the reflecting properties of parabolic mirror s.  Using expansions to see why spherical mirrors have limited ability to concentrate light rays.  Behavior of expansions very close to .  Behavior of expansions far away fr om .

3.04 Taylor's Formula
    

Mathematics.

Taylor's formula for expansions in powers of .

Science and math experience.

Euler, Midpoint and Runge-Kutta approximations of given .    Experiments comparing the quality of Midpoint and Runge-Kutta approximations.  Adaption of Euler, Midpoint and Runge-Kutta approximations to approximati ng the plots of the differential equation , with given.  T aylor's formula in reverse. L'Hospital's rule by dividing the leading term of the expansion of the denominator into the leading term of the expansion of the numerator. Centering the expansion for best approximation.  Experiments comparing the derivative of the expansion and the expansion of  the derivative.

3.05 Barriers to Convergence

Mathematics.

Barriers and complex singularities.  The convergence interval of an expansion as the interval between the barriers.  Why some functions such as have convergence barriers and others s uch as and do not.  Why functions such as and do not have expansions in powers of but do have expansions in powers of for .  Why the convergence intervals for , and are the same.

Science and math experience.

Shortcuts based on the expansion of in powers of .  Using th e expansion of in powers of for drug dosing.  Infinite sum s of numbers resulting from expansions. Barriers resulting from splines. Infinite sums and decimals.  Experiments relating expansions in powers of to interpolating polynomials.  Rung e's disaster.

3.06 Power Series

Mathematics.

Functions defined by a power series.  Functions defined by power series via differential equations.  The power series convergence principle, which s ays that if for some positive number the infinite list
       
is bounded , then the power series
       
converges for .

Science and math experience.

Experiments in trying to plot functions defined by power series.  Experiments in plotting a function defined by a power series via a  differential e quation versus plotting the same function directly through  Mathematica's numerical differential equation solver. The ratio test for  power s eries as a consequence of the power series convergence principle.
The  functions , and from the viewpoint of power series. Experiments in truncation of power series.  The Airy function as a function defined by a power series.

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Converted by Mathematica      January 15, 2001

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