Mathematics 226: study topics

Final exam update

• The Math 200 April 2009 would be a good practice exam, except that we have spent less time on integration and more on the rigorous approach to limits on differentiation, so you could expect 1-2 of the integration questions to be replaced by questions on these topics.
• Similarly for the Math 200 December 2008 exam, except that #2 and one of the integration questions would be replaced.
• On the Math 200 April 2007 exam, #7 and 8 would be simpler, and at least one question (e.g. #1 or 2) would be replaced by a question on limits or differentiation.

Chapter 1, Vectors in Two and Three Dimensions:

• Section 1.1: Vectors in two and three dimensions, basic properties, geometric interpretations
• Section 1.2: Standard basis, parametric equations of lines and curves
• Section 1.3: Dot product, the norm of a vector, angles between vectors.
  • Skip "Vector Projections" and "Vector Proofs", pp. 21-25, except for Proposition 3.4 and its proof. This material should be covered in the co-requisite linear algebra course.
• Section 1.4, "The Cross Product", will be covered in Math 227. The only part of it that will be useful in this course is finding a vector perpendicular to two given vectors in R³. Alternative ways of doing this will be indicated in class.
• Section 1.5: Planes in three dimensions: coordinate equations, parametric equations.
  • Skip "Distance Problems", pp. 44-46
• Section 1.6: skip everything except for vector notation and the dot product in n dimensions. Matrices and matrix operations will be covered in the co-requisite linear algebra course.
• Section 1.7: polar, cylindrical and spherical coordinates.
  • Skip "Standard Bases", pp. 69-71
• Recommended practice problems:
  • Section 1.1: 1-20
  • Section 1.2: 1-39 (pay attention to #26 and 27)
  • Section 1.3: 1-9, 13-16, 18-19
  • Section 1.5: 1-19
  • Section 1.7: 1-35
  • Section 1.9: 2-7, 10-12, 38-39.

Chapter 2: Differentiation in several variables:

• Section 2.1: Functions of several variables
  • Basic definitions: domain, range, injective, surjective
  • Graphs, level and contour curves, sections, level sets in higher dimensions
  • Quadric surfaces: do not try to memorize the names of all surfaces listed on pp. 89-90! The point is for you to learn to graph surfaces given by equations of this type based on investigating their level curves and sections.
• Section 2.2: Limits
  • The rigorous definition of the limit of a function
  • Topology in Rⁿ: open sets, closed sets, boundary, points of accumulation
  • Continuity of functions of several variables
• Section 2.3: The Derivative
  • Partial derivatives
  • Tangent planes and differentiability (Definitions 3.4 and 3.7)
  • Theorems 3.5 and 3.6
  • Please familiarize yourself with the notation concerning derivatives of scalar-valued and vector-valued functions of n variables. However, vector-valued functions and their derivatives (pp. 118-120) will not be a major focus point in this class. They will be used more extensively in Math 227.
• Section 2.4: Higher-order Derivatives
  • Properties of the derivative
  • Higher-order partial derivatives
  • Theorems 4.3 and 4.5
  • Skip "Newton's Method", pp. 130-134
• Section 2.5: The Chain Rule
  • Note that the assumptions in Theorem 5.3 are somewhat weaker than those we used in class (f is assumed to be differentiable at x₀, as opposed to C¹ in its neighbourhood).
  • You should read "Polar/rectangular conversion" (Example 5, pp. 148) and understand how the conversion formulas were derived, but there is no need to memorize them.
• Section 2.6: Directional Derivatives and the Gradient
  • The directional derivative: definition, geometric interpretation
  • Gradients and steepest ascent
  • Tangent planes via the gradient
  • Implicit function theorem and inverse function theorem: you need to remember Theorems 6.5 and 6.7. Theorem 6.6 (the more general implicit function theorem) is used less often and will not be required in this course. Jacobians will be used much more extensively later on, in connection with integration.
• Recommended practice problems:
  • Section 2.1: 1-19, 27, 36-42
  • Section 2.2: 1-23, 28-49
  • Section 2.3: 1-19, 26-32
  • Section 2.4: 1-20, 22
  • Section 2.5: 1-4, 9-14, 22-30
  • Section 2.6: 1-24, 26-39, 40-43, 48-49
  • Section 2.8: 3-9, 18-23, 28-29, 32-34, 36

Chapter 4: Maxima and minima in several variables:

• Section 4.1: Differentials and Taylor's theorem
  • The first order formula and differentials (pp. 233-237)
  • The second order formula and the Hessian (pp. 238-241)
  • Skip "Higher order Taylor polynomials" and "Formulas for remainder terms", pp. 241-244. These will not be used in the sequel.
• Section 4.2: Extrema of functions
  • Critical points
  • "The Hessian Criterion", pp. 247-252, was covered on Wednesday, Dec. 2.
  • Global extrema on compact regions (pp. 252-255).
• Section 4.3: Lagrange multipliers
  • Skip "A Hessian Criterion for Constrained Extrema", pp. 266-270.
• Skip Section 4.4: Some Applications of Extrema.
• Recommended practice problems:
  • Section 4.1: 8-19, 22-27
  • Section 4.2: 3-20 (cover the classification of critical points), 28-35
  • Section 4.3: 2-8, 17-32, 36-37
  • Section 4.6: 5-6, 10-23

Chapter 5: Multiple Integration:

• Section 5.1: Introduction: Areas and Volumes
  • Volumes as iterated integrals
• Section 5.2: Double Integrals
  • Integrals over a rectangle: Riemann sums, integrability conditions (Theorems 2.4 and 2.5)
  • Fubini's theorem
  • Double integrals over general regions: elementary regions, computing double integrals over elementary regions as iterated integrals
• Section 5.3: Changing the Order of Integration
• Section 5.4: Triple Integrals
  • Integrals over rectangular boxes: Riemann sums, integralility conditions (Theorem 4.4), Fubini's Theorem
  • Integrals over general regions: elementary regions, evaluating triple integrals over elementary regions as iterated integrals
• Section 5.5: Change of Variables
  • We are NOT covering change of variables in general. In other words, don't worry about the Jacobians and general change of variable theorems. We are only covering polar, cylindrical and spherical coordinates as indicated below. A derivation of integration formulas specific to these cases has been or will be given in class.
  • Integration in polar coordinates (Examples 10-12, p. 331-333)
  • Integration in cylindrical coordinates (Examples 13-14, p. 337-338)
  • Integration in spherical coordinates (Examples 15-17, p. 339-340)
• Recommended practice problems:
  • Section 5.1: 1-16
  • Section 5.2: 1-16, 20-29
  • Section 5.3: 1-18
  • Section 5.4: 1-25
  • Section 5.5: 13-17, 24-30