# COURSE NOTES

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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

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.

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