The main objective of Calculus III is for students to learn the basics of the calculus of functions of two and three variables. They will study vectors and Euclidean geometry in three-dimensional space, vector valued functions, partial derivatives, the gradient vector, Lagrange multipliers, double and triple integrals and line integrals, culminating with Green's Theorem, Stokes' Theorem and the Gauss Divergence Theorem. They will also apply these ideas to a wide range of problems that include motion in space, optimization, arc length, surface area, volumes and mass. The students should be able to interpret the concepts of Calculus algebraically, graphically and verbally. More generally, the students will improve their ability to think critically, to analyze a problem and solve it using a wide array of tools. These skills will be invaluable to them in whatever path they choose to follow, be it as a mathematics major or in pursuit of a career in one of the other sciences. Students will be required to attend a two-hour laboratory every week, where they will study selected topics. Upon successful completion of the course, students should be able to:
- Find vector and scalar equations of lines and planes in three-dimensional space and apply vector methods to compute distances, angles, areas and volumes.
- Find and interpret partial derivatives, directional derivatives and gradients for functions of several variables.
- Correctly apply the chain rule for transformations.
- Solve unconstrained and constrained optimization problems.
- Set up and evaluate multiple integrals in rectangular, cylindrical and spherical coordinates to find volume, mass and surface area.
- Apply derivatives and integrals to problems of motion and arc length.
- Set up and evaluate line integrals and construct potential functions for conservative vector fields.
- Set up surface integrals and apply the theorems of Green, Stokes and Gauss.
Evaluation of Students
Students will be evaluated on their ability to devise, organize and present complete solutions to problems. While instructors may design their own methods of evaluating student performance, these methods must include in-class examinations, frequent homework assignments and a final exam.
|Topics||Number of Weeks||Sections in Text|
|Vectors and the Geometry of Space||2.5||11.1 - 11.5, 12.1|
|Derivatives and Integrals of Vector-Valued Functions, Arc Length||Lab||11.6 - 11.8|
|Partial Derivatives, Chain Rule and Applications||3.5||12.2 - 12.9|
|Multiple Integration and Applications||4||13.1 - 13.4|
|Triple integrals in cylindrical and spherical coordinates||Lab||13.5|
|Vector fields, line integrals, surface integrals and Theorems of Green, Stokes and Gauss||4||14.1 - 14.8|
Textbooks & Software
Calculus Early Transcendentals Second Edition by Briggs, Cochran and Gillet
Date: August 2017