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Title and hyperlink	Math focus	MAPLE focus area
mk00.mws : Commented and hyperlinked MAPLE index	none	Text and hyperlinks
Part I: Introduction	Calculus, Linear algebra, and Differential Equations	Basic MAPLE skills
mk01.mws : Saying hello	Arithmetic, algebra. In exercise: Integration techniques, leading to discussion of ubiquitous need for keeping track of branch cuts of complex logarithm.	Worksheet format Help, separators (;) Entering commands, Input, output, text etc. Trustworthiness of int() (in exercises).
purefcns.mws : Pure functions	Arrow notation and point of view for functions, operators, procedures, computer programs. Focus: No need for name of "x"-variable	Pure functions
mk04.mws : (Calculus) optimization problems	Calculus: Optimization problems. For which kinds of equations do there exist closed formulas for solutions?	diff(), solve()
mk07.mws : Root locus	Root-locus: Eigenvalues of parameterized curves in the space of 3x3 matrices ONLY UNORDERED set of e-values makes sense!	Symbolic calculations of parameter-dependent eigenvalues. Lots of visualization for "simultaneity".
mk04.mws : Matrix exponentials	Matrix exponentials: Diagonalization, flows of linear DEs	Exact eigenvalue calculations in MAPLE. Micromanaging? Animations of the flows.
why_D.mws : Why D? (as opposed to diff)	The need to distinguish the x in top and bottom of (df(x)/dx) or (df/dx)(x). Use in ICs for IVPs in DE, coordinate changes, tangent maps and the like	Pure functions as opposed to expressions, subs(), variables
vidya_DE.mws : DE example	2nd order linear SINGULAR DE, indicial equations, series solution	Advanced work with dsolve().
Midterm examination	Calculus, Linear algebra, and Differential Equations	Basic MAPLE skills
midsoln.mws : Sample solutions	Taylor expansions and Bernoulli numbers. Jacobian matrix. Evaluating the sine-integral function. Gram-Schmidt orthonormalization. Legendre differential equations.	taylor(),convert(),linalg,jacobian(),fsolve(),dsolve(). Working with matrices and vectors, initial conditions for DEs, various plots, .....
Extended 1st project	Frenet frame animations for parameterized space curves	Combining all basic skills: Calculus, "marrying" linalg and plots packages, procedures,visualization, animation.
torusknot.mws : Torus-knot	Test-curve for the project: smooth, closed curve with nontrivial torsion
mk10.mws : Technical advice	Technical programming advice with deep mathematical roots (noncommuting operations, in common math texts usually tacitly assumed that reader understands the intended order).	Passing parameters by "evaluated expression", delayed evaluation.
Part II: Intro to differential geometry	Curvature, "coordinate changes", invariance under transformations (actions of Euclidean group etc.)	From examples to formal calculations.
arclength.mws : Arclength	Calculating arclength of parameterized curves. Invariance under the action of the Euclidean group.	Limitations of the int() command. Implementing "group action" via explicit parameterization. Proof by direct calculation.
corner.mws : Corners	Standard example of smooth (C^infty) curve that has a corner (distinguish the curve - a function - and its image).
curvature.mws : Curvature	Curvature of plane curves: Arbitrarily parameterized curves and curves parameterized by arc-length. Osculating circles.	Straightforward calculations. Visualization using color nontrivial for curves!). Animations of circles give problems due to scaling.
serret.mws : Integrating the Serret-Frenet-formulas	Integrating the Serret formulas: Curvature (and torsion) completely determines the curve. Time-dependent constant-length loops: Curvature evolving according to a PDE (diffusion equation...)	Closed form symbolic and numeric integration. Visualization and animations.
coords.mws : Linear coordinate change to an adapted normal frame	Orthonormal normal frame of normal vector, and pair of tangent vectors for a surface in 3D.	Purely symbolic calculations involving derivatives of unspecified functions.
euler.mws : Euler's theorem	Euler's theorem for curvatures of intersection curves of smooth surface with normal planes.	Somewhat more complex calculations, but working with specific surfaces. Impressive animations.
gauss.mws : Gauss curvature	Gauss curvature of parameterized surfaces in 3-space.	Straightforward implementation of impressive large formulas for curvature. Visualization using color-coding.
gaussmap.mws : Visualizing the Gauss map	The Gauss map (and the Weingarten map?)	Implementing the Gauss-map, and visualization using side-by-side plots (color keeps track of location). Color alone as attempt to visualize the Weingarten map.
coord0.mws : Normal form for conic sections	Coordinate transformations of functions and equations. Normal forms for conic sections.	Implementing coordinate changes using equations. Assignments can't possibly work!
coord2.mws : Transforming systems of 1st order DEs	Coordinate transformations of systems of differential equations.	Working with derivatives of coordinate transformations, implemented as sets of equations. From specific examples to general formulas.
coord1.mws : Transforming vector fields	Coordinate transformations of vector fields: differential equations, column and row vectors, 1st order partial differential operators differential forms: Tangent maps, pullbacks and push-forwards.	Working with Jacobians. The need to be explicit about changes from (x,y) to (x(t),y(t)) and vice versa. Implementing vector fields as differential operators, and transforming these.... From substitutions in concrete examples with specific formulae to the general case.
coord4.mws : Differential forms and the tensor package	Differential forms, exterior algebra	More details on the D operator. The advanced implementation in the tensor package.
metric.mws : A first intro to Riemannian metrics	A very basic development of Riemannian metrics based on a concrete example, the graph of a function z=f(x,y): Local coordinates, the metric G=g^ij, Christoffel symbols Gamma_ij^k, geodesic equation, visualization.	Handling more complex expressions that lead to large symbolic output: Calculating G for a concrete example as a matrix, calculating its derivatives to form Gamma_ij^k, constructing the geodesic equations, solving them numerically and plotting the solutions.
Part III: Guided student projects	Partial differential equations, special functions	Increasingly formal calculations, combining all MAPLE skills acquired
Ltransfo.mws : Laplace transform, variation of parameters, Green's kernels	Revisit soln techniques for ODEs, comparison, and discuss appropriate notation.	Pushing the limits: From specific examples to general formulas. Some idiosyncratic notation.
drum.mws : Laplace eq, sep of var (not yet complete)	Analyze in detail steps of separation of variables, focus on eigenvalue problem	Release 5 PDEtools package, implement sep of var, incl. eigenvalue problem, by hand.
greens.mws: Intro to Green's functions	Develop idea of fundamental singularity via discrete approximations Focus on ODE BVP.	Advanced book-keeping, some tricky sums and lists, many visuals. Explore the limits of symbolic integration of Green's functions.
	Integrability, Frobenius theorem.