Mathematics 227: study topics

Chapter 1, Vectors in Two and Three Dimensions:

• Section 1.4:
  • The cross product: definition, properties, applications: area of triangle and volume of parallelepiped.
  • (Skip "Torque" and "Rotation of a rigid body", page 33.)

Chapter 3, Vector-Valued Functions:

• Section 3.1:
  • Parametrized curves in Rⁿ, velocity, speed, acceleration, tangent lines.
  • Differentiating vector products (Proposition 1.4, Proposition 1.7).
  • Skip Kepler's laws (Section 3.1, pp. 181-188), except for Propositions 1.4 and 1.7.
• Section 3.2:
  • 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.
• Section 3.3: Vector fields, gradient fields, flow lines.
• Section 3.4:
  • "Del" operator, divergence, curl.
  • Skip "Other Coordinate Formulations", pp. 218-221.
• Recommended practice problems: Section 3.1, 1-10, 15-19, 32; Section 3.2, 1-8, 12-18, 23-33; Section 3.3, 1-12, 17-25; Section 3.4, 1-25.

Chapter 4, Maxima and Minima in Several Variables:

• Section 4.1:
  • Taylor's theorem in several variables: the first order formula (pp. 233-237) was covered in Math 227.
  • Taylor's theorem in several variables: the second order formula, the Hessian (pp. 238-241) Proposition 1.7).
  • Skip "Higher-order Taylor polynomials" and "Formulas for Remainder Terms" (pp. 241-244).
• Section 4.2:
  • Local extrema and critical points in several variables
  • The Hessian criterion for critical points
  • Global extrema on compact regions
• Section 4.3:
  • Lagrange multipliers: one and several constraints
  • Skip "A Hessian Criterion for Constrained Extrema", pp. 266-270.
• Recommended practice problems: Section 4.1, 1-19; Section 4.2, 3-23, 28-47; Section 4.3, 1-10, 16-32.

Chapter 5, Multiple Integration:

• Section 5.1: Areas and volumes
• Section 5.2:
  • Double integrals as limits of Riemann sums
  • Integrability conditions (Theorems 2.4 and 2.5)
  • Fubini's theorem
  • Integration over general regions
• Section 5.3: Changing the order of integration
• Section 5.4: Triple integrals - definition, evaluation, Fubini's theorem
• Section 5.5:
  • Coordinate transformations and their Jacobians
  • Change of variables in double and triple integrals
  • Polar, cylindrical and spherical coordinates were introduced in Math 226. You are expected to be able to use them, including for integration purposes, but we are not covering this as a separate major topic.
• Skip Section 5.6.
• Recommended practice problems: Section 5.1, 1-16; Section 5.2, 1-16, 21-23, 27-29; Section 5.3, 2-18; Section 5.4, 1-25; Section 5.5, 1-19, 23-28; Section 5.8, 1-13, 16-17.

Chapter 6: Line integrals

• Section 6.1:
  • Scalar and vector line integrals: definition and evaluation
  • Reparametrization, orientation of the curve
  • Skip "Numerical evaluation", pp. 375-378.
• Section 6.2: Green's theorem, divergence theorem in 2 dimensions
• Section 6.3: Conservative vector fields
  • Path-independent line integrals
  • Conservative vector fields and gradient fields
  • The curl criterion
  • Finding scalar potentials
• Recommended practice problems: Section 6.1, 1-23; Section 6.2, 1-25; Section 6.3, 1-25

Chapter 7: Surface Integrals and Vector Analysis

• Section 7.1
  • Parametrized surfaces, normal vectors, tangent planes
  • Area of a parametrized surface
• Section 7.2
  • Scalar and vector surface integrals
  • Reparametrization, orientation
• Section 7.3: Stokes's and Gauss's Theorems
• Skip Section 7.4.
• Recommended practice problems: Section 7.1, 1-4, 19-26; Section 7.2, 1-3, 6-23; Section 7.3, 1-16, 19-20; Section 7.5, 22-26.

Chapter 8: Vector Analysis in Higher Dimensions

• Section 8.1: Differential forms, exterior product
• Section 8.2
  • Parametrized manifolds
  • Integrating differential forms over manifolds
  • Special cases: integration over curves and surfaces
  • Skip "Orientation of a paraametrized k-manifold", pp. 489-497
• Section 8.3: Exterior derivative, the generalized Stokes's theorem
• Recommended practice problems: Section 8.1, 1-13; Section 8.2, 6-9, 12-13; Section 8.3, 1, 6-7, 9-10.