Math 450 Exam 2 Study Guide
State the definitions of the following:
Contraction mapping
Contented set
Negligible set
Admissible function
Differential 1-form
The integral of a 1-form along a curve
State the following theorems:
Inverse Mapping Theorem (general multivariate case)
Implicit Mapping Theorem (general multivariate case)
Fubini’s Theorem
Change of Variables Formula (for integrals)
Fundamental Theorem for Path Integrals
Green’s Theorem
Solve problems of the following types:
Find a Taylor Polynomial of a given degree.
Classify critical points as local maxima, local minima, or saddle points.
Verify that a given function is a contraction mapping.
Determine whether a local inverse of a function exists in a neighborhood
of a given point. Find its derivative.
Determine whether an implicit function exists in a neighborhood of a
point on a curve or surface. Find its derivative.
Use Fubini’s Theorem (iterated integrals) to calculate an integral.
Use the change of variables formula to calculate an integral.
Calculate the integral of a 1-form, directly by the definition, using the
Fundamental Theorem for Path Integrals, or using Green’s Theorem.
Otherwise, problems like the assigned exercises.
