COURSE NOTES

Multivariable Calculus

These notes were used to accompany the Hughes-Hallett calculus text, but they are written so as to be largely independent of the text, except of course for differences in notation.

• Quadric Surfaces describes the various quadric surfaces aligned along a coordinate axis.

• Derivatives summarizes the different types of derivatives for scalar functions of more than one variable.

• Differential explains the proper use of the notation of the differential and why it should not be used when doing linear approximation.

• Extrema summarizes the concepts and methods of local and global extrema for functions of two variables.

• Definite Integral Graphic summarizes the components of definite integrals in one, two, or three dimensions.

• Definite Integrals describes the various types of definite integrals, including accumulation over time and aggregation over various spatial domains, including lines, curves, plane regions, surfaces, and volumes.

• Iterated Integrals presents the method for evaluating iterated integrals and general instructions for setting them up.

• Triple Integral Variable Order briefly describes how to choose the most convenient order for setting up triple integrals, based on the characteristics of the region.

• Coordinate Systems is a set of sketches that show how to think of the three common coordinate systems. In particular, a sketch of the rz-plane with radial coordinate ρ and angle of declination φ; shows how to use polar coordinate transformations to connect spherical and cylindrical coordinates.

• Path Integrals presents the concept and evaluation for integrals of scalar functions on curves.

• Line Integrals presents the concept and evaluation for integrals of vector functions along curves.

• Flux Integrals presents the concept and evaluation for integrals of vector functions across surfaces.

Differential Equations

• Euler methods presents the Euler method and Improved Euler methods. This material is taken from my DE book.

• First Order Linear presents the variation of parameters method for solving first order linear equations. This method has some advantages over the usual integrating factor method: it is easy to remember and paves the way for variation of parameters for second-order equations.

• Linear DEs is a comprehensive guide to solving linear differential equations of all orders.

• Undetermined Coefficients presents the method for undetermined coefficients in terms of generalized exponential functions.

• Laplace Transform Overview describes the basic structure of the Laplace transform method.

• Partial Fraction Decomposition presents a complete method for partial fraction decomposition.

• First Translation Theorem explains in detail how to use the theorem that gives the Laplace transform of the product of a transformable function with an exponential function.

• Summary - First Translation Theorem briefly summarizes the previous document.

• Piecewise Continuous Functions introduces the Heaviside function and uses it to obtain single formulas for piecewise-defined functions.

• Second Translation Theorem presents the use of the theorem in two different forms for calculating Laplace transforms of functions with switches and inverting the associated transforms.

Partial Differential Equations

• Green's Function Primer introduces Green's functions using a boundary value problem for an ODE.

• Solving Linear PDEs on Finite Domains generalizes the method of separation of variables so as to include eigenfunction expansion for problems with nonhomogeneous differential equations.

Asymptotic Analysis and Modeling

• Asymptotics Intro introduces asymptotics with examples (8 pages).

• Dominant Balance Example Problem presents the method for using dominant balance arguments to find asymptotic approximations to solutions of nonlinear problems (2 pages).

• Singular Perturbation presents Van Dyke's method for obtaining matched asymptotic expansions for first-order problems, second-order initial value problems, and second-order boundary value problems (9 pages).

• Laplace's Method is a careful presentation of Laplace's method for obtaining asymptotic approximations to definite integrals. Stirling's formula is here, along with relatively simple problems (5 pages).

• Scaling is a study of scaling for a progressively complex set of physical problems, including a continuously stirred tank reactor, a projectile problem, and a model of a damped nonlinear spring (7 pages).

• Steady Flow Across a Flat Plate presents the derivation from first principles of the classic problem for steady flow across a flat plate, sometimes called the Blasius problem (5 pages).

Operations Research

• Queueing Theory Overview provides an overview of the problems and uses of queueing theory.

• Queueing Theory Notes is a self-contained chapter-length treatment of queueing theory, suitable for use as a stand-alone module in an operations research course or as an introduction to a queueing theory course.

• The Augmented Lagrangian Scheme presents the augmented Lagrangian method for solving equality-constrained nonlinear optimization problems. Books on nonlinear optimization sometimes ignore this excellent method.

• Nonlinear Optimization Notes are my hand-written lecture notes for a nonlinear optimization course taught in Spring 2018.