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