# Module MA22S1: Multivariable calculus for science

**Credit weighting (ECTS)**- 5 credits
**Semester/term taught**- Michaelmas term 2013-14
**Contact Hours**- 11 weeks, 3 lectures including tutorials per week
**Lecturer**- Prof. Stephen Britton
**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**- 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.

A collection of useful formulas and facts from the lectures