University of Michigan - Shanghai Jiao Tong University Joint Institute

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Vv 286: Honors Mathematics IV


General information

Credits: No credits are counted towards graduation for those who have completed Vv256
Prerequisites: Vv285 or permission of instructor
Background and Goals: The sequence Honors Mathematics Vv186-285-286 is an introduction to calculus at the
honors level. It differs from the Honors Calculus sequence in that new concepts are often introduced in an abstract
context, so that they can be applied in more general settings later. Most theorems are proven and new ideas are
shown to evolve from previously established theory. Initially, there are fewer applications, as the emphasis is on
first establishing a solid mathematical background before proceeding to the analysis of complex models.The
present course focuses on ordinary differential equations and their applications.
Content: Ordinary differential equations (ODEs) of first order; Systems of first-order equations; the existence and
uniqueness theorem of Picard-Lindeloef; eigenvalue problems, diagonalization and the spectral theorem; Jordan
normal form; application to linear systems of first-order equations; linear second-order equations; elements of
complex analysis and residue theory; the Laplace transform and its inverse with applications to ODEs; power
series solutions of ODEs by the Frobenius method; Bessel’s and Legendre’s differential equations; the Weierstrass
approximation theorem and generalized Fourier series; introduction to the classical partial differential equations
of physics and some basic solutions by separation of variables.
Alternatives: Vv256 (Honors Calculus IV) is an applications-oriented course, which covers much of the same
material.
Syllabus