# Module MA2321: Analysis in Several Real Variables

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2012-13
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 scaler 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
Linear Algebra (MA1212), Analysis (MA1122)
Assessment Detail
This module will be examined jointly with MA2322 in a 3-hour examination in Trinity term, except that those taking just one of the two modules will have a 2 hour examination.