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

Linear Algebra (MA1212), Analysis (MA1124)

Assessment Detail

This module will be examined
in a 2-hour examination in Trinity term.