Syllabus for Several Variable Calculus, Limited Version

Learning outcomes

On the completion of the course the student shall be able to:

• define, identify, explain and exemplify the basic concepts in differential- and integral calculus of several variables;
• account for how the concepts in the previous paragraph are theoretically connected;
• mathematically describe and analyze curves and surfaces in low dimensions;
• calculate derivatives and integrals of functions and vector fields;
• apply the knowledge in previous paragraphs in specific problem solving;
• present mathematical reasoning for others.

Content

Polar, cylindrical and spherical coordinates. Parameterisations of curves and surfaces.

Level curves and level surfaces. Arc length. Scalar and vector valued functions of several variables. Partial derivatives, differentiability, gradient, direction derivative, differential. Derivatives of higher order. The chain rule. Taylor's formula. Optimisation: local and global problems, problems with equality constraints. Multiple integrals, change of variables especially polar coordinates, improper integrals, applications of multiple integrals: volume, centres of mass, etc. Line integrals of vector fields. Green's theorem in the plane. Examples from relevant areas of application.

Instruction

Lectures and problem solving sessions. Team-working may occur.

Assessment

Written examination at the end of the course combined with assignments during the course in accordance with instructions given at the start of the course. 

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.