Math 275 (Analytic Geometry and calculus III)
Office Hours Monday, 10:30am-12:00pm and Thursday 1-3pm at Brink 322; or by appointment email@example.com
Lecture time MWF 9:30-10:20 am, ED 401
Textbook Calculus, by Briggs and Cochran, 11th complete edition
(You may use “volume 2: multivariable” instead of the complete edition. The only difference is the chapter labeling: Chapter 12 in the complete edition corresponds to Chapter 11 in “volume 2: multivariable”, and so forth.)
There will be 10 homework assignments. They will be assigned during the semester on weekly basis, and due at the beginning of Monday's class as specified on the tentative schedule below.
Each student will be given three days of free extension. You can use all the three days
in one assignment, or use them in two or three assignments. After you have consumed your three free extension days, you can still ask for extension, but for each additional extension day,
I will subtract 10% of that assignment.
Midterm Exams (15% X 3)
Three midterm examinations are scheduled on the days specified on the tentative schedule. A make-up examination is possible only if you get my permission before the examination, or with a doctor’s note.
Final Exam (25%)
Final examination is scheduled on Wednesday, May 14, 10:00 am—12:00 pm.
Course grades are determined by: A (90.0%-100%); B (80.0%-89.9%); C (70.0%-79.9%); D (60.0%-69.9%); F (<60.0%).
This course covers multivariable calculus using a highly geometric approach. We first give a treatment of three-dimensional geometry; introduce vectors, dot product, cross product, lines, planes, and how to visualize a surface in three-dimensional space using contour maps and level curves. Then, we cover vector-valued functions (parametric curves), and their derivatives, integrals, curve length, curvature. Next, we extend single variable calculus to multivariable; cover functions of several variables, and their limits and partial derivatives, gradients, directional derivatives, local and absolute maximum/minimum, and Lagrange Multipliers. We then cover two- and three-dimensional integrals in rectangular, polar, cylindrical, and sphere coordinates, and changing variables. Finally, we cover line integrals in two- and three-dimensional spaces, and put the capstone of the course: the Fundamental Theorem of Line Integrals which relates gradients with integrals, and Green’s Theorem which relates line integrals with double integrals. If time permits, we also briefly introduce curl and divergence, surface integrals, Stokes’ Theorem and Gauss’ Theorem.
Course Content Outline
1. Fundamental notions of higher dimensions (Chapters 12 of the textbook)----
• Vector geometry and analytic geometry of R3
• Calculus of vector valued functions with applications to motion in space
• Functions of several variables, including graphs and level curves
2. Differential calculus in higher dimensions (Chapters 13 of the textbook)----
• Differentiation of functions of several variables, including partial derivatives, the chain rule, directional derivatives, gradients, and differentials
• Optimization problems for functions of more than one variable, including critical point analysis and Lagrange multipliers
3. Integral calculus in higher dimensions (Chapters 14 of the textbook)----
• Double integrals in both rectangular and polar coordinates
• Triple integrals in rectangular, cylindrical, and spherical coordinates
• Change of variable theorem
4. Vector calculus o Vector fields (Chapters 15 of the textbook)----
• Line integrals
• Conservative vector fields and the fundamental theorem of line integrals
• Green’s Theorem
• The student will develop the ability to reason geometrically in dimensions higher than two, including mastery of basic notions of vector geometry and equations of lines, planes, and spheres.
• The student will develop the ability to think geometrically about the graphs of functions of two variables, including the significance of partial derivatives, directional derivatives, and gradients.
• The student will learn to do the standard differential operations including partial derivatives and gradients, and how to use them to do simple optimization problems.
• The student will learn to set up and evaluate integrals in two and three variables, including working with any of the standard coordinate systems.
• The student will understand and correctly interpret the basic notions of vector calculus, including line integrals of vector fields and Green’s Theorem.