# M5: Multivariable Calculus - Material for the year 2019-2020

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16 lectures

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.

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.

See the examinable syllabus.

Multiple integrals: Two dimensions. Informal definition and evaluation by repeated integration; example over a rectangle; properties. General domains. Change of variables. Examples. [2]

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

Recap on surface and line integrals. Flux integrals including solid angle. Work integrals and conservative fields. [2]

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

Divergence theorem. Example. Consequences: Green's 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]

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).