Multivariable Calculus · free preview
Parametrized Manifolds and Orientation
k-manifolds as the general shapes of integration, orientation as an ordering of parameters, and piecewise shapes glued with matching sides.
1. The tower of shapes, one definition
The tower of shapes, one definition
2. Orientation as ordered parameters
Orientation as ordered parameters
3. Gluing pieces with matching sides
Gluing pieces with matching sides
4. Worked orientation on the boundary of a square
Worked orientation on the boundary of a square
5. Orienting by permutation
A surface is parametrized by ordered variables . Which reordering gives the same orientation?
6. The seam check
Two surface patches meet along a seam. What does compatible orientation require along it?
7. Dimension census
Which of the following shapes is a parametrized two-manifold?
8. Boundary of a boundary
A compact two-manifold-with-boundary has a boundary that is itself a closed one-manifold. How many boundary curves does a cylinder surface have? Give the count, as a plain number.
9. Orientation count on a connected orientable surface
A connected orientable surface has how many distinct orientations available? Give the count, as a plain number.
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