## Trinity College Dublin, The University of Dublin

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# Module MAU22S01: Multivariable calculus for science

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2021-22
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof Sergey Frolov
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
MAU11S01 & MAU11S02