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Synopsis for Multivariable Calculus


Number of lectures: 16 HT

Course Description

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.

Synopsis

Multiple integrals: Two dimensions. Informal definition and evaluation by repeated integration; example over a rectangle; properties.

General domains. Change of variables.

Jacobian for plane polars; Examples including $ \int_{\mathbb{R}^2}e^{-(x^{2}+y^{2})}\mathrm{d}A. $

Volume integrals: interpretation; change of variable, Jacobians for cylindrical and spherical polars, examples.

Surface integrals. Line integrals. $ \int_{A}^{B}\nabla \phi \cdot \mathrm{d}{\mathbf{r% }} $.

Continuity: Definition of continuity of real valued functions of several variables in terms of limits. $ C^{n} $ functions. Condition for equality of mixed partial derivatives.

Vector differential operators: Scalar and vector fields. Divergence and curl; Calculation, Identities. Higher order derivatives.

Integral Theorems and Applications: Divergence theorem. Example. Consequences: Possibilities: Greens 1st and second theorems. $ \int \int \int_{V}\nabla \phi \, \mathrm{d}V=\int \int_{\partial V}\phi \, \mathrm{d}S $. Uniqueness of solutions of Poisson's equation. Derivation of heat equation. Divergence theorem in plane. Informal proof for plane.

Stokes's theorem. Examples. Consequences.

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).

Last updated by Eamonn Andrew Gaffney on Thu, 07/03/2013 - 7:24am.
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