maple files for
Differential Geometry for Undergraduates

some of these just generate figures for the book, others are for student use (or at least bob in class)

this index file is in progress feb-2013, many more worksheets are available; Exercise numbers may be slightly off if I add extra problems during the semester

Chapter 1.

• linmapvu.mw diagram of how a linear map distorts images in the plane, Figure 1.1 [see also chapter 4 grid addition]
• gl2R-traceinnerproduct.mw trace inner products on 2z2 matrices, Exercise 1.2.1
• asymmatrices-crossprod.mw relation between antisymmetric matrices and double cross product in 3d, Exercise 1.2.3
  rotation-matrices.mw perhaps remove?
• tensortransform.mw transforming the components of a second rank tensor, Exercise 1.3.2
• oneforms.mw visualizing 1-forms, covector addition, Figures 1.4, 1.5
• lincomforback.mw visualizing a change of coordinates, basis, basis parallelopiped, Figure 1.9, reciprocal basis, Appendix
• basischgsr.mw
• basischgde.mw
• basischgde-hw.mw Grid figures for Examples 1.3, 1.4 and Exercises 1.3.2, 1.5.1, 1.5.2, and Figure 3.2.
• basischg-gramschmidt.mw  Gram-Schmidt Example 1.6.?, Exercise 1.6.?
  see also:
  gramschmidt.mw
• eigenvectors_uppertriangular.mw Gram-Schmidt example Exercise 1.6.9
• basischg-secondderivative.mw diagonalize quadratic form Exercise 1.6.10.
• covectorcircleellipse.mw geometric interpretation of index lowering Figures 1.6.?.
• covectorvector.mw relation of covector to its corresponding vector, Figure 1.21, Exercise 1.6.5.
• matrixinnerproducts.mw symmetric/antisymmetric parts, two natural inner products, example 1.6.?
• matrixexponential2by2.mw exponentiating symmetric or antisymmetric 2z2 matrices Exercise 1.7.1
• 2vectorsinr4.mw exercise on 2d subspaces in 4d [move to later chapter?]

Chapter 2.

• multitaylor.mw application of symmetric tensors
• momentsofinertia.mw application of symmetric tensors

Chapter 3.

Chapter 4.

• basischgde-hwchapter4.mw example of changing basis and grid,
  diagram for chapter 1 distortion of figure in plane with new grid
• 2vectorsinr3.mw visualizing the parallelogram of two vectors and their projections onto the coordinate planes in 3d

Chapter 5.

• tangentvectors.mw tangent vectors and vector fields
• fieldplotex.mw flow lines of linear vector fields
• fieldplotex2.mw another example
• angularmomenta.mw the rotation vector fields at a common point on sphere
• paracylindrical.mw paracylindrical coordinates in plane example
• polargrid.mw polar coordinate grid in coordinate and physical space
• polarcoords-metric.mw derivation of metric in polar coordinates
• polarcoordtriangle.mw diagram of polar coordinate triangle, dual map of rotation by 90 degrees
• rotationmatrixeigenvectors.mw eigenvector analysis of an antisymmetric matrix generating a rotation

Chapter 6.

Chapter 7.

Chapter 8.

• tensortemplate.mw [to use for a new Riemannian metrix]
• tensortemplatelorentz.mw [to use for a new Lorentz metric]
• tensor_package_test.mw [many examples]
• rotationgroup-eulerangles.mw

Chapter 9.

Chapter 10.

• extrinsicsaddle.mw extrinsic curvature for the parabolic hyperboloid graph, intrinsic curvature

Chapter 11

• threesphere.mw as an example of Stokes Theorem (Gauss): integrate a radial 1-form over the sphere, equal to the integral of its differential, which is 4 times the volume 4-form. see the induced orientation (opposite sign to 3D intuition).

Appendix A

• A.1,2
  hyperbolic.mw [hyperbolic-trigonometric.mw figures for this appendix as well as rotations of the plane]
• hypaddidentites.mw pseudorotations derived from the addition formulas for the hyperbolic sine and cosine
• specialrelativity.mw
• matrixinnerproducts-nullrotations.mw null rotations etc from Lorentz group
  A.3
• helixTNB.mw T-N-B vectors along helix in R^3
• helix-lorentz.mw analog in 3d Minkowski spacetime, timelike curve:
  lorentzhelix.mw circular orbit in a plane, world line in 3-d Minkowski spacetime
• uniformmagneticfieldhelix.mw in 4-d Minkowski spacetime
  A.4
• futurepseudoshere.mw unit future hyperboloid in 3d Minkowski spacetime
• twinparadoxfamily.mw the family of pseudocircles passing through two points on the time axis, related to the twin paradox and time dilation

Appendix ?

• tensor_package_test.mw many examples using Maple tensor package to calculate connection and curvature
• tensortemplate.mw template to use to calculate new examples, Riemannian, including 2-surface extrinsic curvature
• tensortemplatelorentz.mw template to use to calculate new examples, Lorentz, including 2-surface extrinsic curvature
• cavatappi helical tubelike 2-surfaces, Euclidean and Lorentz examples to link

3-feb-2013