MAS1006: Advanced Calculus

Lectures: 9am Tuesdays, 12pm Wednesdays, 9am Thursdays LT6 for 12 weeks, starting Tuesday 30 January 2001.

Lecturer: Robin Chapman, room 811, rjc@maths.ex.ac.uk, web page:

http://www.maths.ex.ac.uk/˜ rjc/rjc.html ,

course web page:

http://www.maths.ex.ac.uk/˜ rjc/courses/ac00/ac00.html .

Syllabus: Ordinary differential equations: Dependent and independent variables, order, degree, linearity. General and particular solutions. Solution of firstorder equations: separable, homogeneous, linear. Linear equations with constant coefficients: auxiliary equation, complementary function and particular integrals, method of undetermined coefficients. Linear equations with variable coefficients: Euler’s equation. Simultaneous equations. Sequences and series: Limits, L’Hôpital’s rule. Sequences: convergence, divergence, sequential definitions of limits of functions and continuity. Infinite series. Sums of arithmetic and geometric progressions. Integral, comparison, limit comparison, ratio tests. Absolute and conditional convergence, alternating series. Taylor and Maclaurin series, radius of convergence, differentiation and integration of series. Partial differentiation: Functions of several variables. Contour lines and sections. Partial derivatives, first and higher order. Linearization, total derivatives. Exact differential equations. Chain rule for functions of several variables. Implicit differentiation. Directional derivatives. Tangents and normals. Maxima and minima: Taylor’s theorem for several variables. Maxima and minima for several variables. Lagrange multipliers. Multiple integrals: Double and triple integrals. Interpretation via area and volume. Curvilinear coordinate system: Cartesian, polar, spherical polar, cylindrical polar. Transformation of multiple integrals and the Jacobian formula.

Recommended Book: Thomas & Finney Calculus (9th ed.).

Course Work: Six example sheets due in fortnightly from 5th February 2001, three MAPLE assignments. Test (25%): 1205-1255, Wednesday 25th April, 2001. Examination (75%): 3 hours, May/June 2001.