School of Mathematics
Module MA22S1 - Multivariable calculus for science
2011-12 (SF Science
SF Human Genetics
SF Physics & Chemistry of Advanced Materials
SF Medicinal Chemistry
SF Chemistry with Molecular Modelling
)
Lecturer: Prof. M. Vlasenko
Requirements/prerequisites:
prerequisite: MA11S2
Duration: Michaelmas term, 11 weeks
Number of lectures per week: 3 lectures plus 1 tutorial per week
Assessment:
ECTS credits: 5
End-of-year Examination:
2 hour examination in Trinity term.
Description:
As its name suggests, multivariable calculus is the extension of
calculus to more than one variable. That is, in single variable
calculus one studies functions of a single independent variable
y=f(x). In multivariable calculus we will study functions of two
or more independent variables z=f(x, y), w=f(x, y, z), etc. These
functions are essential for describing the physical world since
many things depend on more than one independent variable. For
example, in thermodynamics pressure depends on volume and temperature,
in electricity and magnetism the magnetic and electric fields are
functions of the three space variables (x,y,z) and one time variable
t.
Multivariable calculus is a highly geometric subject. We will relate
graphs of functions to derivatives and integrals and see that
visualization of graphs is harder but more rewarding and useful in
several geometric dimensions. By the end of the module you will
know how to differentiate and integrate functions of several
variables.
As the Newton-Leibniz rule relates derivatives to integrals
in single variable calculus, in multivariable calculus this is done
by the three major theorems (Green's, Stokes' and Gauss').
These are considered in MA22S2 in the second term.
Syllabus
- Areas and Lengths in Polar Coordinates
- Lines and Planes, Cylinders and Quadric Surfaces
- Vector-Valued Functions and Space Curves
- 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 Polar Coordinates
- Triple Integrals in Cylindrical and Spherical Coordinates
- Change of Variables, Jacobians
- Vector Fields and Line Integrals
Recommended reading:
Calculus, H. Anton, I. Bivens, S. Davis.
Learning
Outcomes:
On successful completion of this module, students will be able to:
- write equations of planes, lines and quadric surfaces in 3-space
- determine the type of a conic section and write a 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 parametric curve;
compute the length of a portion of a curve
- apply the 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 the 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 of change of variables
- use integrals to find quantities defined via integration
(such as average, area, volume, mass).
Nov 11, 2011
File translated from
TEX
by
TTH,
version 3.89.
On 11 Nov 2011, 15:34.