Prerequisites: Math 154

Credit: (4-0)4

Content: Introductory Topology : Domains and regions. Functions of several variables. Limits and continuity. Partial derivatives. Directional derivatives. Gradients. Differentials and the tangent plane: the fundamental lemma, approximations. The mean value, the implicit and the inverse function theorems. Extreme values. Introduction to vector differential calculus: the gradient, divergence and curl. Curvilinear coordinates.

Goals: A continuation of Math 153, Math 154. Differential calculus in several variables is rigorously developed.

Course Outline:

(2 Weeks) Introductory topology : Balls, accumulation points, open, closed sets with geometric examples; concepts of boundary, interior, and closure;  continuity, connectedness and compactness.

(2 Weeks) Functions of several variables: Examples of functions ƒ : Rⁿ → R and Rⁿ → R^m.  Sketching domains and graphs of some functions. Definition of the limit and continuity of functions, examples on approaching to a point P from various directions, geometric interpretation of continuous surfaces.

(1 Week) Partial derivatives: Definition and geometric meaning, relation to existence of limit, continuity, higher order partial derivatives, directional derivatives, the gradient.

(4 Weeks) The differential and the tangent plane: Definition of the differential and its relation to partial derivatives and continuity, examples of functions w/o differential, the geometric meaning of the differential: Existence of the tangent plane. The fundamental lemma, total derivative formula and the chain rule, Jacobian, general differential, differential approximation for functions ƒ : Rⁿ → R^k.

(1 Week)The Mean Value Theorem, Implicit Function and Inverse Function Theorems.

(2 Weeks)Extrema of functions of several variables: Hessian, Lagrange Multipliers with applications.

(1 Week) Introduction to vector differential calculus: Gradient, divergence, curl, orthogonal curvilinear coordinates, Laplacian in cylindrical and spherical coordinates.

Grading:  There will be two midterm exams and a final exam,

Midterm 1:  35% ,  November 10,  2016 at 17:40 Exam Results  Exam  Places   (You may see your exam papers on November 23, 2016 at M-201 during office hours)

Midterm 2:  35%,    December 14,  2016 at 17:40 Exam Results  Exam  Places 

Final:  50%,     January 16, 2016, at 17:00 Exam Results   Exam Places

Total is  120 (20% bonus)

Letter grades will be given according to catalog!

Make-up: , 2016 at  (Make-up exam will be from all topics!)  

Attendance: I will take 24 random attendance during the term, to enter the final exam you should be present in at least 18 of them.

Teaching Assistant: Örsan Kılıçer 

                               Office Hours: Wednesdays 15:30-17:30 at Z-44

Exercises:

Week 1

Week 2

Week 3

Week 5-6

Week 7-8-9-10

Exercise sheet