Advanced Multivariate Calculus
The
course covers functions, differentiation and integration in n-dimensional Euclidean space. We discuss the basic algebraic, Euclidean, and topological structure of n-dimensional
space. We then introduce the notion of continuous and differentiable
functions between such spaces. Fundamental properties such as the
inverse and implicit functions theorems will be explained. This is
followed by the notion of the Hessian of a function and (constrained)
critical point theory - minima/maxima/saddles. The 2nd part of the
course treats differentials, integration over curves and surfaces, and
the Stokes Theorem (including special case incarnations such as the
Green and Gauss Theorems).
Grading
Home work problems will be assigned on a regular basis and graded.
There will be a midterm exam and a final exam.
The total grade will be the equally weighted average of those three
grades. D is in the range of 60-70, C 70-80, B 80-90, and A 90-100.
Grader
Feifei Xie, LGRT, 1423 D
xie@math.umass.edu
By appointment
Text
J. Marsden and A. Tromba, Vector Calculus (any edition is fine)
H. M. Schey, div, grad, curl and all that
M. Spivak, Calculus on Manifolds
• hw 1 • hw 2 • hw 3 • hw 4 • hw 5 • hw 6 • midterm • hw 7 • hw 8 • hw 9 • final
Course log
Chapters refer to Marsden and Tromba's "Vector Calculus" 6th edition.
Week 1: chapter 2.1 (review of Calc I, II, III and Linear Algebra) and beginning of 2.2 (structure of Rn)
Week 2: chapter 2.2 and beginning of chapter 2.3
Week 3: chapter 2.3 and beginning of chapter 2.5
Week 4: chapter 2.5 and beginning of chapter 2.6
Week 5: chapter 2.6, 1.4 (coordinates), parts of 3.5 (inverse and implicit function theorems)
Week 6: submanifolds parametrized and equations defined; tangent space
Week 7: Min/Max theory, critical points: chapters 3.2, 3.3
Week 8: Level sets, gradients: chapter 2.6
Week 9: Hessian, 2nd derivative test, gloabal max/min, Lagrange multipliers: chapters 3.3, 3.4
Week10: Vector fields, curves, line intergrals, gradient vector fields: chapter 4
Week11: Integrals and Stokes Theorem