My mathematical skills and interests lie in:
Markov semigroups
Spectral theory and functional calculus
Random dynamics
Markov chain Monte Carlo methods
Hypocoercivity
Stochastic homogenization
Camrud - A Second Course in Calculus - Dynamics in Disguise.pdfThis book should serve as a Calculus II text for a traditional undergraduate path. Dynamics, as a tool of applied math, is highlighted as a motivating reason to study the myriad of subjects that Calculus II covers. The text also emphasizes the duality between (differentiable) functions as the primary objects of study in (continuous) calculus, and sequences as the primary objects of study in discrete calculus.
Topics covered include:
Refreshers on arithmetic, operations, trigonometry, exponentials, and logarithms
Complex arithmetic and operations
Refreshers on the (continuous) derivative
Refreshers on the (continuous) integral, including integration by substitution
Integration by parts
Sequences
Finite differences
Sums (as the "discrete integral")
Common rules of sums
Continuous and discrete dynamics
Initial value problems
Separable ODEs
Asymptotics: Improper integrals and infinite series
Parametric curves as descriptions of 2-dimensional dynamics
Polar curves
Partial fraction decomposition
Trigonometric substitution
Convergence tests for improper integral and infinite series
The Taylor series
Applications of power series
Arc lengths and areas of parametric curves
Surface areas and volumes of revolution
Furthermore, video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).
Ordinary Differential Equations Done RightThis book should serve as a second course in ODEs for a traditional undergraduate path, or as a first course for advanced undergraduates and early graduate students. Most notably, a full sequence of calculus and linear algebra are necessary prerequisites.
Topics covered include:
Complex numbers and functions
Linear algebra (including Jordan form, matrix functions, and the spectral mapping theorem)
Infinite-dimensional vector spaces, function spaces, and inner product spaces
The Fourier series, Fourier and Laplace transforms
The matrix exponential solution to systems of linear ODEs
Duhamel's principle for nonhomogeneous linear ODEs
Variation of parameters for nonhomogeneous linear ODEs
Nonlinear maps, and linearization
Closed-form solutions to nonlinear ODEs (exact equations and integrating factors)
Euler, Runge-Kutta, and Galerkin methods for approximating solutions
Chaos theory, and the Hartman-Grobman theorem for stability of equilibrium points
The Lie series, and an introduction to PDEs
Both proofs-based, and purely computational exercises are included, with the intent that the course could be administed with-or-without a formal proofs course as a prerequisite.
Furthermore, video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).
Camrud - Partial Differential Equations Done Right.pdfThis book should serve as a first course in PDEs for a traditional undergraduate path. Most notably, a full sequence of calculus, linear algebra, and ODEs are necessary prerequisites.
Topics covered include:
Construction of PDEs from physical principles
Domains and boundaries in arbitrary dimension
A review of ODEs solution methods
Outer products and tensor fields
The Taylor series for scalar fields
The Fourier series and transform for scalar fields
Flows, Hamiltonians, and the method of characteristics
Construction of the Laplace, heat, and wave equations
Separation of variables
Fourier methods for the solution to the Laplace, heat, and wave equations
Classification of elliptic, parabolic, hyperbolic PDEs
Integral transforms, the Dirac delta function, and Green's functions
Eigenvalues and eigenfunctions
The spectral theorem for Hermitian PDE operators
The Cauchy-Kovalevskaya theorem and power series methods
Finite difference and Galerkin methods
Video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).
Department of Mathematics & Statistics, Middlebury College, Middlebury, VT
Institutional email: ecamrud [at] middlebury [dot] edu
Personal email: evancamrud [at] gmail [dot] com
Ars in omnibus est. "Art is in everything." I took Latin in high school, which makes me 5x more likely to accidentally summon demons as an adult. But in all seriousness, I like to find art in everything, and such a notion defines my approach to mathematics, as well as life.
I am currently a lecturer and visiting research scholar at Middlebury College. I study nonlinear stochastic differential equations, statistical sampling algorithms, and related problems in functional analysis.
I reside with my wife Kira Rahn and our adorable cat Jillie Bean.
When I am not researching or teaching, you will most often find me hiking, where I enjoy foraging for wild fruits. I also enjoy cooking/baking, drinking at the local breweries (should I be advertising that?), and playing tabletop games.