Syllabus for Math 2561
Honors Multivariable Calculus
Spring 2016
The timing of topics in this syllabus is approximate
and may change during the semester.
Week 1 (Lectures on 1/12, 1/14):
Text: Shifrin Chapter 2
Concepts: Scalar- and vector-valued functions of several variables. Level sets and graphs.
Open and closed sets in Rn.
Least upper bound property of R.
Convergence of sequences in Rn.
Limits and continuity for vector-valued functions.
Skills: Recognize and give examples of vector-valued functions of physical and engineering relevance.
Determine convergence and limits of simple sequences.
Determine whether specific functions are continuous.
Write simple proofs about open and closed sets, convergence, and continuity.
Weeks 2-3 (Lectures on 1/19 to 1/29):
Text: Shifrin Chapter 3
Concepts: Partial, directional, and total derivative.
The total derivitive as best linear approximation to the function.
Chain rule. The gradient. Curves and curvature. Kepler's laws.
HIgher order partial derivatives.
Skills: Compute partial, directional, and total derivatives.
Compute linear approximations to a function.
Interpret derivatives in terms of level sets and graphs.
Compute arc lengths. Write simple proofs involving derivatives.
Weeks 4-5 (Lectures on 2/2 to 2/11):
Text: Shifrin Chapter 5, sections 1-4
Concepts: Continuous functions on compact sets achieve a maximum.
Critical points and maxima for multivariable scalar functions.
The second derivative test. Constrained optimization and Lagrange multipliers.
Skills: Solve multivariable optimization problems.
Write simple proofs about compactness and extrema.
Week 6 (Lectures on 2/16, 2/18):
Weeks 7-8 (Lectures on 2/23 to 3/3):
Text: Shifrin Chapter 4, Section 5, and Chapter 6.
Concepts: Implicit, explicit, and parametric descriptions of a subset of Rn.
Contractions and fixed points.
Inverse and implicit function theorems.
Introduction to manifolds.
Skills: Translate between implicit, explicit, and parametric descriptions of simple sets. Analyze Newton's method in simple examples.
Find values and derivatives of implicitly defined functions.
Give simple examples of manifolds.
Write simple proofs involving implicitly defined functions
Weeks 9-10 (Lectures on 3/8 to 3/17):
Shifrin Chapter 7
Concepts: Multiple integrals, iterated integrals, and Fubini's theorem.
Polar, cylindrical, and spherical coordinates.
The Jacobian determinant and change of coordinates.
Applications.
Skills: Compute multiple and iterated integrals.
Changes coordinates and/or change the order of integration to simplify problems.
Apply integration to solve problems in mechanics, gravitation, and electrostatics.
Write simple proofs involving integration.
Skills: Compute with differential forms (i.e., compute and
simplify wedge products, exterior derivatives, and pull-backs).
Week 12 (Lectures on 4/5, 4/7):
Weeks 13-15 (Lectures on 4/12 to 4/26):
Text: Shifrin Chapter 8, Sections 3-7
Concepts: Fundamental theorem for line integrals.
Line integrals and Green's theorem.
Surface integrals and Gauss's theorem.
Classical Stokes' theorem. General Stokes' theorem.
Applications to physics (and maybe topology).
Skills: Compute line, surface, and solid integrals, directly and using the theorems.
Connect "div, grad, curl and all that" to the exterior derivative.
Develop some intuition for vector fields with div or curl zero.
Final Exam on Tuesday May 3rd from 6:00pm to
8:50pm covering material up to and including week 15