Module MA2321: Analysis in Several Real Variables

Credit weighting (ECTS)

5 credits

Semester/term taught

Michaelmas term 2014-15

Contact Hours

11 weeks, 3 lectures including tutorials per week

Lecturer

Prof. David Simms

Learning Outcomes

On successful completion of this module, students will be able to:

Prove the chain rule for functions defined on finite dimensional real vector spaces;
Prove the inverse function theorem for functions defined on finite dimensional real vector spaces;
Prove the implicit function for functions defined on finite dimensional real vector spaces;
Define smooth manifolds, tangent spaces, vector fields, 1-forms, push-forward of tangent spaces and pull-back of 1-forms;
Define the differential of a scalar field, show that the differentials of coordinates are dual to the partial derivatives, and show that the differential commutes with the pull-back;

Module Content

Derivative as a linear operator, partial derivatives, C1 functions are differentiable, equality of mixed partials, inverse function theorem, implicit function theorem, smooth manifolds, tangent spaces, vector fields, 1-forms, push forward of tangent spaces and pull-back of 1-forms, diferentials.

Module Prerequisite