M2AA2 Multivariable Calculus

Lecturer: R. Craster

Syllabus

This is the Mathematics concerned with extending the basic concepts of differentiation and integration (elementary calculus) to functions of many variables. This is a sketch of the syllabus and a basic plan of action. Functions from R^m to R^n: Define a scalar function, a vector, directional derivatives (connect to elementary vectors) and then introduce a Jacobian. Use this to define continuity and differentiability. implicit function theorem, inverse function theorem, (this will be done without proving either, statement plus examples and consequences) higher derivatives. Vector calculus: vector fields, grad, div, curl, surface integrals, divergence theorem, Stokes' theorem, curvilinear coordinates. Integrals in Rn: curves, line integrals, Green's theorem, transformation of integrals (connects back to Jacobians etc). Partial differential equations [simple wave, Laplace, heat]. Concentrate upon Laplace. Cartesian tensors: simple properties and notation. [Previous year's examinations have included calculus of variations, method of images and separation of variables these are not examinable this year as they were either not covered or not covered until very late in the course.]

Contact details Website for course: http://www.ma.ic.ac.uk/ ~ rvcras/M2AA2/course.html
Here are example sheets and courseworks ( all sheets have solutions online). Solutions will be added as we go through the course.

Email address: r.craster and then (@imperial.ac.uk), for anything to do with this course put M2AA2 in the subject heading.

Office hour: Friday 16:00 in 644