Mathematics 227: study topics

Chapter 3, Vector-Valued Functions:

• Parametrized curves in Rⁿ, velocity, speed, acceleration, tangent lines.
• Differentiating vector products: Proposition 1.4, Section 3.1.
• Paths on spheres: Proposition 1.7, Section 3.1.
• Skip Kepler's laws (Section 3.1, pp. 181-188), except for Propositions 1.4 and 1.7.
• Length of a path, arclength parametrization.
• Unit tangent vector, curvature. Computing curvature: from the definition, using formula (17) in Section 3.2.
• The moving frame: unit tangent, principal normal, and binormal vectors. Osculating plane, torsion.
• Tangential and normal components of acceleration.
• Vector fields, gradient fields, flow lines.
• "Del" operator, divergence, curl. Skip "Other Coordinate Formulations", pp. 218-221.

Chapter 5, Multiple Integration:

• Review Sections 5.1-5.4, see Math 226 topics.
• Coordinate transformations, Jacobian, change of variables in definite integrals.
• Definite integrals in polar, spherical, cylindrical coordinates.
• Improper integrals (see Section 5.6, problems 23-33).
• Applications: areas and volumes, average value, center of mass, centroid (we will focus on examples where calculations are reasonably short). Skip "moments of inertia", pp. 353-355.

Chapter 6, Line Integrals:

• Scalar and vector line integrals.
• Green's Theorem, Divergence Theorem. Applications: using line integrals to compute areas, converting line integrals to double integrals and vice versa.
• Conservative vector fields, gradient fields and potentials. The "curl criterion" on simply connected regions.

Chapter 7, Surface Integrals:

• Parametrized surfaces, normal vectors, tangent planes.
• Computing areas of parametrized surfaces.
• Scalar and vector surface integrals.
• Orientable vs. non-orientable surfaces. (We skipped the analytic proofs of Theorems 2.4 and 2.5 (Addendum to Section 7.2), instead relying on the fact that the geometrical meaning of the integrals is independent of parametrization except for orientation.)
• Stokes's Theorem and Gauss's Theorem, with applications. The interpretation of divergence and curl.

Chapter 8, Manifolds and Integrals of k-forms:

• Differential k-forms, exterior product.
• Integration of 1-forms over curves.
• Integration of 2-forms over surfaces.
• Integration of k-forms over k-manifolds. This and the next three topics will not be covered in detail. We will discuss the main concepts and their geometrical meaning, but will not attempt to prove theorems or work out examples.
• Orientation of a parametrized manifold.
• Exterior derivative of a differential form.
• Generalized Stokes's Theorem.