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

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.