MAU22S01 Multi-variable calculus for science
| Module Code | MAU22S01 |
|---|---|
| Module Title | Multi-variable calculus for science |
| Semester taught | Semester 1 |
| ECTS Credits | 5 |
| Module Lecturer | Prof. Manya Sahni |
| Module Prerequisites | MAU11S02 Mathematics for scientists II |
Assessment Details
- This module is examined in a 2-hour examination at the end of Semester 1.
- Continuous assessment contributes 20% towards the overall mark.
- Any failed components are reassessed, if necessary, by an exam in the reassessment session.
Contact Hours
11 weeks of teaching with 3 lectures and 1 tutorial per week.
Learning Outcomes
On successful completion of this module, students will be able to
- Determine the equations of planes, lines and quadric surfaces in R3.
- Recognise the equations of conic sections and determine the change of coordinates that turns a given quadratic equation into its standard form.
- Use cylindrical and spherical coordinate systems.
- Determine the tangent line, unit tangent vector, normal and binormal vectors as well as the curvature of a parametic curve at a given point.
- Use integrals to compute the length of a portion of a parametric curve.
- Employ the above techniques to describe the motion of a particle in space.
- Calculate limits and partial derivatives for functions of several variables.
- Find the local linear and quadratic approximations of a function of several variables and write the equation of the plane which is tangent to its graph at a given point.
- Compute directional derivatives and use the gradient vector to find the direction of most rapid increase for a function of several variables.
- Use Lagrange multipliers to find the local maxima and minima of a given function.
- Compute double and triple integrals using either Fubini's theorem or a change of variables.
- Use integrals to compute physical quantities such as average, area, volume and 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.
- Gradient vector and directional derivatives.
- 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 and Jacobians.
Recommended Reading
- Calculus: Late transcendentals by Anton, Bivens and Davis.