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
- 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.
- Vector fields, line, surface and volume integrals.
- 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.
- Orthogonal curvilinear coordinates; length of line element; grad, div and curl in curvilinear coordinates; spherical and cylindrical polar coordinates as examples.
- A first look at Legendre polynomials.
- Fourier series: full, half and arbitrary range series. Parseval's Theorem.
- Laplace's equation. Uniqueness under suitable boundary conditions. Separation of variables. Two-dimensional solutions in Cartesian and polar coordinates. Axisymmetric spherical harmonic solutions.
Learning outcomes
http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5102
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)
Undergraduate Modules for Academic Year 2012–13 (last updated 4 March 2013)
