1 Functions

1.1 What is a function?

    1.1.1 What is a function? Informal Discussion,

    1.1.2 Example: Height vs time of a free falling particle,     

    1.1.3 Linear relations,

    1.1.4 Functions given by data tables,

    1.1.5 Table of functions (These two sections utilize the material due to 

                                    Professor J. Uhl)

    1.1.6 Domain and range of function

1.2 Elementary functions: Informal review

    1.2.1 Elementary functions: monomials,

    1.2.2 Elementary functions: trigonometric functions,

    1.2.3 Elementary functions: exponential and

                logarithmic functions

1.3 Function as map

    1.3.1 Function as map: summary,

    1.3.2 What is a function?  Short History,

    1.3.3 Jumps and cusps

1.4 Translation and reflection of functions

    1.4.1 Translation of functions,

    1.4.2 Reflection --- Mirror images,

    1.4.3 Example: J S Bach, Die Kunst

                der Fuge BWV 1080

1.5 Scaling and self-similarity

    1.5.1 Scaling of functions, homogeneous functions,    

    1.5.2 Self-similar functions, fractal curves,

    1.5.3 Brownian motion (Wiener process)

1.6 Inverse function  -- folding along  y = x

1.7 Periodic functions

1.8 Composition of functions

1.9 Arithmetic of functions

Minimum requirements

(1) Clearly understand what a function is.

(2) Elementary functions must be understood intuitively (for example, you must be able to sketch graphs of representative functions: rigonometric functions, polynomials, exponential and logarithmic functions, hyperbolic functions).

(3) Given a function, be able to express  its translation and  reflection in terms of the original function.  Also clearly understand advanced and retarded phase shift.

(4) What should you do if you wish to expand (magnify) a function in the x or y direction?

(5) When is the inverse function definable?

Key Words

function (variables, domain, range), traveling wave, phase shift (advanced, retarded), translation, reflection, inverse function, periodic function, composition of functions

2 Polynomials

2.1 Qualitative features of polynomials

    2.1.1 Monomials and polynomials: definition,

    2.1.2 Qualitative features of monomials, 

    2.1.3 Qualitative features of polynomials, 

    2.1.4 Highest order term determines global behavior of polynomials,     

    2.1.5 Landau's O

2.2 Polynomial curve fitting

    2.2.1 Heuristic Weierstrass, 

    2.2.2 Bernstein polynomials

2.3 Derivative of polynomials

    2.3.1 Growth rate of polynomials, 

    2.3.2 Expression for growth, 

    2.3.3 Linearity of growth rate, 

    2.3.4 Derivative of  monomials,

    2.3.5 Binomial theorem and another calculation

            of derivative of monomials,

    2.3.6 Rudiments of combinatorics, 

    2.3.7 Landau's o, 

    2.3.8 Derivative of polynomials

2.4 Rational functions

    2.4.1 Rational functions: definition, 

    2.4.2 Poles --- blowing-up, 

    2.4.3 Global behavior of rational functions

Minimum requirements

(1) You must be able to sketch the graph of monomials and polynomials.

(2) Which is larger xn or xm if n < m?  Understand the significance of the degree of a polynomial.

(3) Differentiation is to compute the slope of  the tangent line.  Be able to differentiate polynomials (be sure). Memorize the formulas.

(4) What is linearity?

(5) Practice how to use Landau's o.

(6) Continuous functions can be fit by a polynomial as accurately as you wish.

(7)  Be able to give asymptotic behavior of rational functions.

Key Words

monomial, polynomial (degree), linear combination, linear operation, Landau's O, Landau's o, polynomial fitting (Weierstrass' theorem), binomial theorem, rational function, pole

3 Powers

3.1 Elementary Review of power function

3.2 Construction of general power function

3.3 Derivative of power functions  (rational exponents)

3.4 Rational and Irrational Numbers

    3.4.1 Rational numbers

    3.4.2 Not all numbers are rational --- Grecian surprise,

    3.4.3 Irrational numbers,

    3.4.4 Rational numbers are 'as many as' integers,

    3.4.5  Which are more numerous, irrational numbers

                or rational numbers?

    3.4.6 Irrational numbers are far more numerous

                than rational numbers,

    3.4.7 Rational numbers are dense.

Minimum requirements

(1) Be sure to handle elementary calculations with ease.

(2) Be able to sketch power functions for indices less than or larger than 1.

(3) It is desirable to understand how various power functions are defined starting with monomials.

(4) Understand 1/(1 + x) = 1 | x + o[x].

(5) Be able to differentiate power functions.

(6) Be able to explain what is rational and what is irrational.

[(7) Irrational numbers are uncountable; rational numbers are countable but dense;

Key Words

power function, rational number, irrational number, [dense, countable, uncountable, diagonal argument ]

4 Continuous Functions ---

            Intuitive Approach

4.1 Minimum Summary

    4.1.1  Magnifying graphs ---

            local structure of functions,

    4.1.2 Continuous functions ---

            intuitive characterization,

    4.1.3 Continuous function ---

            idea for more precise characterization,

    4.1.4  Continuity at a point --- definition

4.2 Intermediate value theorem

4.3 Maximum and minimum

4.4 Convexity

    4.4.1 Convex functions,

    4.4.2 Convex functions are continuous

4.5 Don't believe continuous curves are always docile

Minimum requirements

(1) Be sure to be able to state continuity precisely in terms of the ∂ and δ.  You must also be able to explain the idea to your intelligent but uninitiated friend.

(2) Itemize consequences of continuity (intermediate value theorem, maximum value theorem).

(3) What is a convex function?

Key Words

continuity, intermediate value theorem, (absolute) maximum, minimum, convexity, Jensen's inequality  

5 Exponential and

       Logarithmic Functions

5.1 Exponential Functions

    5.1.1 Definition of exponential functions,

    5.1.2 Qualitative behavior of exponential functions,

    5.1.3 Functional relations of exponential functions,

                summary

5.2 Exponential Growth and Decay

    5.2.1 Scaling property of exponential functions,

    5.2.2 Exponential growth and decay: examples

5.3 Logarithm

    5.3.1 Logarithm: inverse of exponential functions, 

    5.3.2 Properties of logarithmic functions,

    5.3.3 Base change of logarithmic functions

5.4 e

    5.4.1 Exponential function for small variables,

    5.4.2 There must be a number e,

    5.4.3 Natural logarithm

5.5 Derivatives of exponential and logarithmic functions

    5.5.1 Derivative of ex,

    5.5.2 Derivative of logarithms

Minimum requirements

(1) Be able to sketch exponential function ax for various a.

(2) Have a clear image of exponential growth or decay.

(3) Be sure to have no problem with changing bases of logarithmic functions.

(4) It is advantageous to memorize log102 and log103 (see here).

(5) Be able to explain the significance of e and natural logarithm.

(6) Be able to differentiate exponential and logarithmic functions. The formulas must be memorized.

(7) Be able to explain lim (1+ x/n)n = ex.  Memorize this result.

(8) Understand ex = 1 + x + o[x], log(1 + x) = x + o[x]. Memorize this result.

Key Words

exponential function, exponential growth (decay), half life, logarithm, e, natural logarithm

6 Trigonometric Functions

6.1 Trigonometric Functions

    6.1.1 Radian,

    6.1.2 Sine and cosine,

    6.1.3 Addition formulas and De Moivre's formula,

    6.1.4 Tangent,

    6.1.5 Reciprocal trigonometric functions

6.2 Derivative of trigonometric functions

6.3 Euler's formula (informal)

    6.3.1 Review of complex numbers,

    6.3.2 Euler's formula,

    6.3.3 Differentiation with the aid of Euler's formula

6.4 Inverse trigonometric functions

    6.4.1 Inverse sine = Arcsin,

    6.4.2 Derivative of  Arcsin,

    6.4.3 Inverse cos = Arccos,

    6.4.5 Inverse tan = Arctan,

    6.4.6 Derivative of  Arctan, 

    6.4.7 Other inverse trigonometric functions

Minimum requirements

(1) Clearly remember the definitions of trigonometric functions, esp., sin and cos.  Be able to sketch the graphs of sin, cos, tan, sec and cosec. 

(2) Then, memorize representative values (check here).  Be familiar with the practical way to relate sin and cos.

(3) Understand how to derive addition formulas via rotation.

(4) Not only be able to differentiate trigonometric functions, but also be able to explain why the formulas are right (Understand sin θ > θ, cos θ > 1). Then, memorize the derivatives of sin, cos, and tan.

(5) Understand the relation between de Moivre's formula and Euler's formula. Then, memorize Euler's formula. Be able to write trigonometric functions in terms of complex exponentials.

Key Words

radian, cos, sin, tan, rotation, addition formula, de Moivre's formula, double-angle formula, half-angle formula, sec, cosec, cot, Euler's formula,  complex  exponential, Arcsin, Arccos,  Arctan, 

7 Hyperbolic Functions

7.1 Hyperbolic Functions

    7.1.1 Sinh and cosh,

    7.1.2 Addition formulas,

    7.1.3 Hyperbolic tangent, 

    7.1.4 Reciprocal hyperbolic functions

7.2 Derivative of hyperbolic functions

7.3 Inverse hyperbolic functions

    7.3.1 Inverse sinh = sinh-1 = Arcsinh,

    7.3.2 Derivative of  sinh-1,

    7.3.3 Inverse tanh = tanh-1 = arctan,

    7.3.4 Derivative of  arctanh,

    7.3.5 Other inverse hyperbolic functions

Minimum requirements

(1) Memorize the definitions of sinh, cosh, and tanh in terms of e±x.  You must be able to sketch their graphs. 

(2) Understand their relation to trigonometric functions.  It is advantageous to memorize cosh ix = cos x and sinh ix = i sin x.  Addition formulas etc can be obtained easily from the corresponding trig formulas.

(3) Be able to differentiate hyperbolic functions; you must be able to explain the formulas, and then memorize derivatives of sinh, cosh, and tanh.

(4) Recall that inverse hyperbolic functions have forms in terms of exponential and logarithmic functions.

Key Words

sinh, cosh, tanh, addition formula  

8 Global Properties of

     Elementary Functions

8.1 Rational functions revisited

8.2 Exponential growth is always faster than polynomials

            (eventually)

8.3 Logarithmic growth rate is slower than

               any positive power

Minimum requirements

1) Global asymptotic behavior of rational functions must be easily estimated.

(2) Exponential growth is always faster than any power growth (check). You must be able to explain why this is so.

(3) Logarithmic growth is always slower than any power growth.

9 Practical

        Differentiation

9.1 Differentiation

    9.1.1 Differentiation --

            an informal definition,     

    9.1.2 Tangent line of a function at a point,    

    9.1.3 Linear Operation,    

    9.1.4 Differentiation is a linear operation

9.2 Basic Differentiation Rules

    9.2.1 Derivative and linear response,

    9.2.2 Difference symbol,

    9.2.3 Differentiation of product,

    9.2.4 Chain rule

9.3 Derivative of inverse functions

9.4 Logarithmic derivative

9.5 Newton's method

9.6 Higher order derivatives

    9.6.1 Second order derivative,

    9.6.2 Higher order derivatives,   

    9.6.3 Leibniz' rule

9.7 Partial derivative

Minimum requirements

(1) Memorize the definition of derivative in terms of  limit. You must be able to tell its intuitive meaning. Understand the difference between differentiability and mere continuity.

(2) Be familiar with the use of Landau's o to express differentiability of a function at a given point.  Especially understand differential and the use of difference symbol.

(3) Given a graph of a function, you must be able to sketch its derivative.

(4) Check your memory of derivatives of elementary functions.

(5) Where do derivatives show up in the actual contexts? 

(6) Be familiar with the use of derivatives to describe the change.

(7) Differentiation is a linear operation. Derivatives describe linear responses.

(8) Elementary rules of differentiation must be memorized, but you must be able to explain why they hold with the aid of differential (for example, product rule).

(9) Be able to differentiate inverse functions. Sometimes it is advantageous to remember the formulas for inverse trigonometric and hyperbolic functions.

(10) Logarithmic differentiation may facilitate computing the zeros of derivatives.

(11) You must be able to write down the local form of twice differentiable functions.

(12) Given a graph of a function, you must be able to sketch its second derivative.

(13) Understand the logic of mathematical induction.

(14) Remember the definition of partial derivatives.

[(15) Understand the principle of Newton's method, and be familiar with its use.]

Key Words

differentiation, differential, tangent line, dimension, operator, superposition principle, linear response, product rule, chain rule, logarithmic derivative, higher order derivatives,  convexity, Leibniz rule, mathematical induction, partial differentiation, partial derivative [Newton's method, nonlinear difference equation (map), iteration, fixed point]

velocity, acceleration, heat capacity, susceptibility,

10 Curve Sketching

10.1 Maximum and minimum: extremum

10.2. If differentiable f is extremal,  f ' vanishes

10.3 Rolle's theorem

10.4 Mean value theorem

10.5 Cauchy's mean value theorem

        and l'Hospital's rules

10.6 Taylor's formula

    10.6.1 Polynomials and higher order derivatives,

    10.6.2 Taylor's formula

10.7 f ' = 0 everywhere implies f is constant

10.8 f ' ρ 0 everywhere implies f is monotone

        increasing

10.9 f '' > 0 implies convexity of f

10.10 Inflection point

10.11 Local quadratic ap

Minimum requirements

(1) Understand the possible implications of  f ' = 0. Geometrically clearly understand Rolle's theorem. 

(2) Geometrically understand the mean value theorem.

(3) Understand Taylor's formula, and memorize it.  It is worth memorizing Taylor series expansions of ex, sin x, etc.

(4) f ' ≥ 0 everywhere implies that f  is monotone increasing.

(5) f " ≥ 0 implies convexity.

(6) Understand the possible implications of f" = 0.

(7) To summarize what we can learn from the first and the second derivatives the local quadratic form is useful.  Be sure to understand its implication.

(8) You must be able to semiquantitatively sketch the graph of a function with the aid of its first and second derivatives.

Key Words

extremum (maximum, minimum), Rolle's theorem, mean value theorem, Cauchy's mean value theorem, Taylor's formula, Taylor expansion (Taylor series),  monotone increasing, inflection point, local quadratic approximation,

Minimum requirements

(1) Review rudiments of rational and irrational numbers. Clearly recognize that the usual elementary exposition of rational and irrational numbers is incomplete.

(2) At least understand the outline of Dedekind's theory of real numbers.

(3) Understand the concept of convergence.

(4) Bounded monotone sequences converge.

(5) Cauchy sequence is equivalent to converging sequence.

(6) \[Sum]xn/n! = ex.  Be able to explain this converging sequence and memorize this.

[(7) Understand the definition of  real numbers in terms of cuts.

(8) Inequalities, and arithmetic of real numbers must be defined to be consistent with that for rational numbers.

(9) Understand the meaning of the continuity of reals in terms of cut of  real numbers.

(10) Be able to explain the equivalence of real numbers and decimal numbers.

(11) Bounded sets have infimum and supremum.

(12) Nested closed intervals share a unique point.

(13) Bounded infinite point sets have an accumulation point = Bolzano-Weierstrass's theorem.

(14) A bounded closed set is compact.

(15) Continuity of real numbers can be characterized by three axioms.]

Key Words

real number, convergence, convergent sequence, bounded monotone sequence, Cauchy sequence, series,

[Dedekind's cut, cut of real numbers, supremum, infimum, nested closed interval sequence, accumulation point, Bolzano-Weierstrass' theorem, Heine-Borel's covering theorem, compact set, axiom of continuity (Dedekind's axiom, Cantor's axiom, Weierstrass' axiom)] 

12 Continuity

12.1 Continuity

    12.1.1 Limit of function,

    12.1.2 Convergence of function sequence,

    12.1.3 Continuity of function,

    12.1.4 Continuity and uniform convergence

12.2 Intermediate Value Theorem

12.3 Maximum Value Theorem

12.4 Uniform Continuity

12.5 Weierstrass' Approximation Theorem

Minimum requirements

(1) Definitions of convergence of function values, of function sequence, etc., should be clearly understood and memorized.

(2) Uniformly converging continuous function sequence defines a continuous function.

(3) Understand the proof of the intermediate value theorem.

(4) Understand the proof of the maximum value theorem.

(5) The range of a continuous function on a closed finite domain is closed.

Key Words

limit,continuity, left limit, Cauchy's criterion, sequential continuity, pointwise convergence,  uniform convergence, intermediate value theorem, maximum value theorem, uniform continuity

13 Differentiation

13.1 Differentiation

    13.1.1  Definition of derivative,

    13.1.2  Left and right derivatives

13.2  Summary of elementary rules

13.3 Theorems of derivatives

        (Again, but this time rigorously)

    13.3.1 Rolle's theorem,

    13.3.2  Mean-value theorem,

    13.3.3  Applications of mean-value theorem,

    13.3.4 Cauchy's mean-value theorem

13.4 Taylor's formula

    13.4.1 Higher order derivatives revisited,

    13.4.2 Taylor's formula revisited,

    13.4.3 Real analytic function

Minimum requirements

(1) Derivative geometrically understood as a slope is not a perfect understanding of differentiation; differentiation is a more subtle procedure than computing the slope.

(2) Review elementary rules. Use the check quiz.

(3) Be sure to understand and then memorize the logic used in the proofs of basic theorems of derivatives (Rolle's theorem, mean value theorem).

(4) Understand the derivation of Taylor's formula, and memorize it.

Key Words

definition of derivative, left (right) derivative, Taylor's formula, remainder, real analytic function,

14 Ordinary Differential Equations

         --- Preview

14.1 What is a differential equation?

    14.1.1  Differential equations,

    14.1.2  Auxiliary conditions

14.2 Simple differential equations I First order

    14.2.1 Simple differential equations whose

                solutions we already know,

    14.2.2 Simple Applications

14.3 Euler's method -- numerical approach

    14.3.1 Euler's basic idea,

    14.3.2 Illustrative examples of Euler's

                    method,

    14.3.3 How to use Mathematica for simple

                    differential equations

14.4 How to qualitatively understand

            differential equations.

14.5 Simple differential equations II Second

            order

14.5.1 Harmonic oscillators, 14.5.2 Phase flow approach

Minimum requirements

(1) What is (ordinary) differential equation?

(2) Understand why auxiliary conditions are needed (see also @)

(3) Understand the relation between a parameter family of curves and a differential equation.

(4) Understand the general solutions of simple ODE (ordinary differential equation) dx/dt = αx, dx/dt = αx + β, dx/dt = x/t) and memorize the results.

(5) Understand the logic to demonstrate the uniqueness of the solution.

(6) Be familiar with the use of simple ODE through representative examples such as chemical reactions.  Understand how to derive the logistic equation.

(7) What will you do if you cannot find solutions that have neat analytic expressions?

(8) Understand the principle of Euler's method to solve ODE numerically.

(9) Understand how to guess the general behavior with the aid of the vector field.

(10) Understand harmonic oscillator as a representative example of second order differential equations.  Memorize its general solution.

(11) Understand how to sketch the vector field on the phase space and how to qualitatively understand the second order equation with the aid of the phase portrait.

Key Words

differential equation, order, auxiliary condition, general vs particular solution, integration constant, Newton revolution, uniqueness, radioactive decay, half life, logistic equation, qualtitative approach,  Euler's method, vector field, direction field,  autonomous vs non-autonomous, harmonic oscillator, damped harmonic oscillator, phase space, phase portrait, van der Pol equation,

15 Integration --- Basic

15.1 Primitive function

15.2 Integral

15.3 Riemann integral

15.4 Riemann integral as signed area

15.5 Riemann sum and elementary

            integration

15.6 Changing integration variables

15.7 Integration by parts

15.8  Whose primitive functions can we find

            analytically?

    15.8.1 Elementary facts to be

                remembered,

    15.8.2 What integration skills do we need?

    15.8.3 Seemingly benign but

                hard-to-integrate functions

15.9  Integration of rational functions

    15.9.1 Partial fraction expansion of

                rational functions,

    15.9.2 Integration with the aid of

                partial fraction expansion

15.10  Integration of rational functions of

            sin and cos

    15.10.1 Polynomial of sin and cos,

    15.10.2 Rational functions of sin and cos,

    15.10.3 Trig substitution

15.11  Improper integrals

    15.11.1 Unbounded integration range,    

    15.11.2 Integration of singular functions,

    15.11.3 General properties of

            improper integrals

Minimum requirements

(1) What is integration?  Understand the relation to differentiation. (Summary)

(2) Integration may be understood in terms of a linear operator.

(3) Understand the indefinite integration formulas for elementary functions, and memorize them.  Be able to do simple integrals quickly (check this and this).

(4) Definite integral can be compute if we know indefinite integrals; however, do not forget the warning.

(5) Understand Riemann's idea on the Riemann integrals.

(6) Understand Riemann integral as signed area. Understand integration geometrically (backward integral, joining the integration range, etc.)  For example, you must be able to visualize the fundamental theorem of calculus. (Summary)

(7) Intuitively understand |∧ f(x) dx| ≤ ∧ | f (x)|dx.

(8) Do not have any difficulty in changing the integration variables.

(9) Be familiar with integration by parts (check).

(10) Basically, you need not be able to do analytic integration very skillfully, but must be able to clearly recognize the doable integrals (basically, rational functions, and rational functions of trigonometric functions, polynomial times trigonometric functions, quadratic irrational functions,  some mixtures of exponential and logarithmic functions and polynomials) as such (and why).  However, also remember seemingly easy integrals cannot be done analytically.

(11)  To memorize basic analytic integration techniques is advantageous; of course you must be able to explain why the techniques work. 

(12) When the integration range is  not finite and/or the integrand is not bounded, the integral must be defined appropriately with the aid of limiting procedures, and the resultant integral is called an improper integral.

Key Words

primitive function, integration, integral, definite integral, integrable, integrand, fundamental theorem of calculus, indefinite integral, integration operator (see also here), backward integration, Riemann sum, Riemann integral, signed area, integration by parts, partial fraction expansion, improper integral, gamma function, beta function,  

[Lebesgue integral]

16 Practical Laplace

            Transformation

16.1 Introduction to Laplace Transformation

    16.1.1 Definition and elemantary

    16.1.2 Fundamental one-to-one

    16.1.3 Basic properties

16.2 Constant coefficient linear ODE with

            the aid of  Laplace transformation

    16.2.1 Linear ODE with constant

                    coefficients

    16.2.2 Linear ODE with constant

            coefficients: general theory with the aid

            of Laplace transformation

    16.2.3 How to perform inverse Laplace

            transform

    16.2.4 Constant coefficient linear ODE:

            summary

    16.2.5 Green's function for ODE: a preview

16.3 Actual examples

Minimum requirements

(1) Memorize the definition of  Laplace transformation.  It is a linear transformation.

(2) Differentiation corresponds to multiplication of s, so linear constant coefficient ODE become algebraic equations.

(3) Laplace  transformation is one to one, so we can recover the original function from its Laplace transform.

(4) To invert Laplace transforms we can use lookup tables. 

(5) However, the Laplace transforms of, constants, exponential functions,  sin x, cos x, sinh x and cosh x are worth memorizing. 

(6) Also some general rules, multiplicative exponential factor, time shift, and convolution in particular should be remembered. (i.e, what is the effect of multiplying s, multiplying x, etc.?)

(7)  The general solution to the homogeneous equation + a particular solution to the inhomogeneous equation = the general solution to the inhomogeneous equation.

(8) Second order equation should be completely understood with the aid of characteristic roots. Also realize that if the second order equation is understood, higher oder equations are similarly understood.

(9) Understand the general behavior of relaxation systems.

(10) Understand the transient and steady behaviors under external forcing of (damped) oscillators intuitively (i.e.,  you should be able to guess their general behavior).

(11) You must be able to produce several concrete examples in which constant coefficient linear ODEs appear.

[(12) Green's function = response to impact (impulse) can be used to construct particular solutions to inhomogenous equations. Delta function is a generalized function.]

Key Words

Laplace transformation, Laplace transform, inverse Laplace transformation, homogeneous equation, inhomogeneous equation, constant coefficient linear ODE, scaling of variable, convolution, characteristic polynomial, characteristic root, general structure of solution to linear ODE, resonance, relaxation, relaxation time, transient behavior,      

[Green's function, delta function, generalized function, impulse  ]

17 Ordinary

    Differential

        Equations

17.1 Elementary Review of ODE

17.1.1 What is the ODE??

17.1.2 Solution to ODE

17.1.3 Qualitative understanding of ODE

17.2 How to derive ODE

17.3 Elementary Analytic Methods

17.3.1 Separation of variables

17.3.2 Homogeneous equations

17.3.3 Perfect differential equations and integrating factor

17.3.4 First order linear ODE

17.3.5 Special named equations

17.4 17.4 Fundamental Problems of ODE

17.4.1 What are fundamental questions?

17.4.2 Existence

17.4.3 Uniqueness

17.4.4 Extension

17.4.5 Well-posedness

17.5 Linear ODE

17.5.1 Review of constant coefficient linear ODE

17.5.2 17.5.2 General theory of linear ODE

17.5.3 General constant coefficient liner ODE

Minimum requirements

(1) n-th order ODE can be written as a first order ODE of n dependent variables.

(2) Review the classification of solutions.

(3) Review qualitative ways to understand ODE.

(4) You must be able to derive simple ODEs.  Separating each mechanism of time evolution is a practical way of constructing an ODE model.

(5) You must be able to use the method of separation of variables for simpler cases. To this end you should be familiar with primitive functions of representaitve elementary functions. Be familar with the calculation in terms of differentials.

(6) The concept of integrating factor must be understood.

(7) You must be able to solve first order linear ODE; also remember the method of variation of coefficients.

(8) There are named ODE that may be solved with certain tricks.  Although you must understand why such tricks work, but there is almost no need of memorizing these special cases.  When you encounter such equations, first go to your favorite stylebook.

(9) Memorize the concept of envelop, envelop curves, and their relation to special solutions.

(10) What are the most fundamental questions of ODE?  You must be able to itemize them and explain why they are important.  (There are ODEs with non-unique solutions.)

(11) Understand the Lipschitz condition geometrically.

(12) Under the Lipschitz condition unique existence of local solutions can be demonstrated (Cauchy's unique existence theorem). Usually,  local solutions are extended to its maximal existence time interval.  The obtained global soutions are continuously dependent on the initial conditions (i.e., well- posedness).  It is important to understand the concept of well-posedness.

(13) If the maximal existence interval is inside the domain of the ODE, the solution reaches the boundary of the domain (usually it explodes). 

[(14) Gronwall's inequality is a very useful inequality worth remembering.]

(15) Understand the general structure of the solution to matrix linear ODE.

(16) Understand the general theory of second order linear ODE.  

(17) Understand how to construct the general solution to constant matrix equation in terms of exponential or matrices.   A practical method to compute exponential of matrices should be followed at least once with a simple example.  It is useful to remember all the cases for 2×2.

(18) Itemize what we can learn from the phase portrait.

(19) Understand how to classify the fixed point; the concept of hyperbolicity is important.  Prior to this of course you must understand various concepts of stability.  Again you should understand them in terms of 2×2 systems.

(20) linear stability ⇒ asymptotic stability ⇒ stability.  Note that ⇐ is untrue.

(21) Be able to analyze the stability of a fixed point of a general ODE.  Understand the importance of structural stability.

(22) Intuitively understand the idea of Lyapunov to study the stability of a fixed point.

23. (23)Remember the concept of limit cycles, and the related bifurcation.  Roughly remember what chaos is.
    

Key Words

n-th order ODE, normal form, autonomous vs. nonautonomous, linear vs nonlinear equation, solution (curve), particular solution, singular solution, general solution, van der Pol equation, vector field, phase portrait, limit cicycle, fixed point and its stability, Voterra's equation, homogeneous ODE, perfect differential form, integrating factor, variation of coefficients, envelop, existence problem, uniqueness problem, extension problem, well-posedness, Peano's existence theorem, Lipschitz condition, Picard's approximation method, Cauchy's unique existence theorem, maximal existence interval, Gronwall' inequality, explosion, global extendability,  matrix ODE,  exponential of matrix, fundamental set of solution, evolution operator = fundamental matrix, Wronskian, exp(A), sink, saddle, source, hyperbolic fixed point, stability, asymptotic stability, linear stability, linearized equation, structural stability, stable and unstable set, Lyapunov function (in the narrow sense),  ω-limit set,  limit cycle, Poincare-Bendixson's theorem, Hopf bifurcation,