# Vector Calculus

Course category:
2nd year
Module code:
MAT00012I
Year:
2011/12
Term:
Spring
Term:
Summer
Credits:
20
Lecturer:
Professor Paul Busch
Lecturer:
Dr Carl Chalk

## Aims

Part A:

• To deepen and extend the Stage 1 “Calculus” module, with a more mature look at the fundamental concepts of infinitesimal calculus from the viewpoint of vector-valued functions of many variables.
• To discuss the three essential ingredients of calculus—continuity, differentiability, and integrability—bringing out the distinctive flavour of each theory, and describing their inter-relationships and applications.
• To gain a deeper understanding of the three famous differential operators of classical vector calculus: div, grad and curl.

Part B:

• To describe the differential calculus of scalar and vector fields in R3.
• To extend the theory of integration to volume integrals and integrals over surfaces in R3.
• To describe the theorems of Stokes and Gauss which link these topics together, and thereby to complete the chain of generalisations of the Fundamental Theorem of Calculus encountered previously.
• To determine when a potential function and vector potential exist, and calculate them.

## Learning objectives

At the end of the module you should be able to

Part A:

• Appreciate the concept of continuity for functions of many variables, and recognize a good supply of continuous functions.
• Understand the concept of linear approximation, recognize a wide range of differentiable functions of many variables, and differentiate them.
• Appreciate, and use, the chain rule.
• Integrate vector and scalar fields along contours in n-dimensional spaces.
• Discern when a multi-dimensional vector field is conservative, and construct a potential.
• Find a harmonic conjugate for a harmonic function in the plane.
• Calculate the flux of a vector field through a planar domain.
• Express a line integral as a double integral, and vice versa.

Part B:

• Compute integrals over a variety of regions of space.
• Parametrise a variety of surfaces in space and compute surface integrals of vector fields.
• Use index notation to manipulate vector expressions.
• Work with the fundamental differential operators of vector calculus.
• Understand the relationship between volume integrals and surface integrals, and surface integrals and line integrals.
• Use the integral theorems to move from one type of integral to another.
• Apply the techniques of vector calculus to a range of problems from geometry and physics.
• Determine when a potential function and a vector potential exist, and calculate them.

## Syllabus

PART A

Introduction to vector calculus.  Elementary topology of Euclidean space.

Limits and continuity.  Basic properties of vector limits.

Multivariable differentiation.  Linear maps. Composition of continuous functions. Linear approximation and the multivariable derivative. Basic properties of the derivative. The Chain Rule. Paths, directional derivatives, partial derivatives and the Jacobian matrix. The differentiability test.

Line integrals.  Line integrals and mechanical work. Fundamental theorems of line integration. Poincare's Lemma, curl, work and energy. Line integration with respect to arc length.

Green's Theorem.  Harmonic conjugates, Green's Theorem. Change of variables formula for double integrals (revision from Calculus). Examples of Green's Theorem, application to area.

Divergence Theorem in two dimensions.

Symmetry principles for double integrals.

PART B

Review of vector algebra.

Index notation.  Kronecker delta and epsilon, and their properties.

Vector differential operators in 3 dimensional Euclidean space.  Gradient, divergence, curl, directional derivatives and the Laplacian; differentiation of products; chain rules. (Partly a review of objects introduced in Calculus.) Div, grad and Laplacian in curvilinear coordinates; solution by separation of variables; physical examples (eg. vibrations of a drumskin).

Parametric description of surfaces in space, and their orientation.

Surface and volume integrals.  Integration over a parametric surface and volume integrals, including Cartesian, spherical polar, and cylindrical polar coordinates.

Stokes and Gauss theorems, and their applications.

Intrinsic definitions of gradient, divergence and curl. (if time permits)

Poincare's lemma.  Scalar and vector potentials.(if time permits)

## Recommended texts

• KF Riley, MP Hobson, SJ Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, CUP 2006 (3rd ed)
• H F Davis & A D Snider, Vector Analysis, Allyn & Bacon.

## Teaching

• 3 lectures and 1 seminar per week in Weeks 2-10 of the Spring term and Weeks 1-3 of the Summer term
• 2 revision classes Week 4 of the Summer term.

## Assessment

Three hour examination summer term.

## Elective information

a mature study of fundamental ideas in advanced calculus.

Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.