MATH 412 (Advanced Calculus with Applications)
| DESCRIPTION |
Rigorous discussion of fundamental concepts of analysis in
several
variables combined with computational algorithms such as Newton's
method
and the method of steepest descent. Application to problems in
many
areas with a view to both computing solutions and deriving qualitative
conclusions about the models. (This course is not open to students who
have completed Math 350 and 351. Credit may not granted for both Math
412
and 411.) |
| PREREQUISITES |
Calculus I, II, III, Linear algebra, one semester of advanced
calculus
in one variable (MATH 410) |
| TOPICS |
The basics
Vector norms on Rn
Open sets
Closed sets
Compactness
Connectedness
Continuous functions
Max and min
Uniform continuity
Differentiable functions (linear approximation)
Mean value theorem
Hessian matrix
Positive definite matrices
Second derivative test
Taylor expansions for functions of several variables
Solving equations
Matrix norms
Perturbations of invertible liner maps
Contraction mapping principle
Inverse function theorem
Newton's method
Implicit function theorem
Optimization
Method of steepest descent
Constrained optimization: method of Lagrange
multipliers
Kuhn-Tucker formulation of inequality constraints
Integration in several variables
Extensions of trapezoid and Simpson's rule to higher
dimensions
Change of variable in multiple integrals
Applications of change of variable in numerical
calculation and statistics
Derivation of the Euler equations of fluid flow
|
| TEXT |
Text(s)
typically used in this course. |
|
