# M5: Multivariable Calculus (2018-2019)

## Primary tabs

2018-2019
Lecturer(s):
Dr Richard Earl
Course Term:
Hilary
Course Lecture Information:

16 lectures

Course Overview:

In these lectures, students will be introduced to multi-dimensional vector calculus. They will be shown how to evaluate volume, surface and line integrals in three dimensions and how they are related via the Divergence Theorem and Stokes' Theorem - these are in essence higher dimensional versions of the Fundamental Theorem of Calculus.

Learning Outcomes:

Students will be able to perform calculations involving div, grad and curl, including appreciating their meanings physically and proving important identities. They will further have a geometric appreciation of three-dimensional space sufficient
to calculate standard and non-standard line, surface and volume integrals. In later integral theorems they will see deep relationships involving the differential operators.

Course Syllabus:
Course Synopsis:

Multiple integrals: Two dimensions. Informal definition and evaluation by repeated integration; example over a rectangle; properties. General domains. Change of variables. Examples. [2.5]

Volume integrals: Jacobians for cylindrical and spherical polars, examples. [1.5]

Recap on surface integrals. Flux integrals including solid angle. [1.5]

Scalar and vector fields. Vector differential operators: divergence and curl; physical interpretation. Calculation. Identities. [2.5]

Divergence theorem. Example. Consequences: Greens 1st and second theorems. $\int_V \nabla \phi dV = \int_{\delta V} \phi dS$. Uniqueness of solutions of Poisson's equation. Derivation of heat equation. Divergence theorem in plane. Informal proof for plane. 

Stokes's theorem. Examples. Consequences. The existence of potential for a conservative force. 

Gauss' Flux Theorem. Examples. Equivalence with Poisson's equation.