Modules

MTH5102 Calculus III

Description

The module develops the elements of vector calculus and advanced topics in ordinary and partial differential equations, such as special functions, Fourier series and Laplace's equation, for application in subsequent applied mathematics modules.

Details

Organiser: Dr W J Sutherland

Level: 5 Credit value: 15 Semester: A

Overlaps:

• MAS204 Calculus III
• PHY122 Mathematical Techniques 2

Essential prerequisites:

• MTH4101 Calculus II

and

• MTH4103 Geometry I

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~wjs/MTH5102

Syllabus

1. Arc-length of plane curves: length of a parametric curve, length of a curve y = f(x) . Length of the circumference of a circle, ellipse. Area and length in polar coordinates.
2. Vector fields, line, surface and volume integrals.
3. Grad, div and curl operators in Cartesian coordinates. Grad, div, and curl of products etc. Vector and scalar forms of divergence and Stokes's theorems. Conservative fields: equivalence to curl-free and existence of scalar potential. Green's theorem in the plane.
4. Orthogonal curvilinear coordinates; length of line element; grad, div and curl in curvilinear coordinates; spherical and cylindrical polar coordinates as examples.
5. A first look at Legendre polynomials.
6. Fourier series: full, half and arbitrary range series. Parseval's Theorem.
7. Laplace's equation. Uniqueness under suitable boundary conditions. Separation of variables. Two-dimensional solutions in Cartesian and polar coordinates. Axisymmetric spherical harmonic solutions.

Learning resources

Main text:

• Thomas' Calculus, 11th Edition (Addison Wesley)

Other texts:

• M R Spiegel, Vector Analysis (Schaum Outline Series, McGraw-Hill)
• S Simons, Vector Analysis for Mathematicians, Scientists & Engineers (Pergamon Press)