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Module MA22S1: Multivariable calculus for science

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2015-16
Contact Hours
11 weeks, 3 lectures including tutorials per week
Prof John Stalker
Learning Outcomes
On successful completion of this module, students will be able to:
  • Write equations of planes, lines and quadric surfaces in the 3-space;
  • Determine the type of conic section and write change of coordinates turning a quadratic equation into its standard form;
  • Use cylindrical and spherical coordinate systems;
  • Write equations of a tangent line, compute unit tangent, normal and binormal vectors and curvature at a given point on a parametic curve; compute the length of a portion of a curve;
  • Apply above concepts to describe motion of a particle in the space;
  • Calculate limits and partial derivatives of functions of several variables
  • Write local linear and quadratic approximations of a function of several variables, write equation of the plane tangent to its graph at a given point;
  • Compute directional derivatives and determine the direction of maximal growth of a function using its gradient vector;
  • Use the method of Lagrange multipliers to find local maxima and minima of a function;
  • Compute double and triple integrals by application of Fubini's theorem or use change of variables;
  • Use integrals to find quantities defined via integration in a number of contexts (such as average, area, volume, mass)
Module Content
  • Vector-Valued Functions and Space Curves;
  • Polar, Cylindrical and Spherical Coordinates;
  • Quadric Surfaces and Their Plane Sections;
  • Functions of Several Variables, Partial Derivatives;
  • Tangent Planes and Linear Approximations;
  • Directional Derivatives and the Gradient Vector;
  • Maxima and Minima, Lagrange Multipliers;
  • Double Integrals Over Rectangles and over General Regions
  • Double Integrals in Cylindrical and Spherical Coordinates;
  • Triple Integrals in Cylindrical and Spherical Coordinates;
  • Change of Variables, Jacobians
Module Prerequisite
MA1S11 & MA1S12
Recommended Reading
Calculus. Late trancendentals. by H.Anton, I.Bivens, S. Davies
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination. Supplemental exams if required will consist of 100% exam.