M5: Multivariable Calculus - Material for the year 2018-2019

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. [4]

Stokes's theorem. Examples. Consequences. The existence of potential for a conservative force. [2]

Gauss' Flux Theorem. Examples. Equivalence with Poisson's equation. [2]

Reading List: 

1) D. W. Jordan & P. Smith, Mathematical Techniques (Oxford University Press, 3rd Edition, 2003).

2) Erwin Kreyszig, Advanced Engineering Mathematics (Wiley, 8th Edition, 1999).

3) D. E. Bourne & P. C. Kendall, Vector Analysis and Cartesian Tensors (Stanley Thornes, 1992).

4) David Acheson, From Calculus to Chaos: An Introduction to Dynamics (Oxford University Press, 1997).