Category: Maths

1. FUNCTIONS OF A SINGLE REAL VARIABLE – Limits and continuity – Complete discussion includes proofs of all theorems in the body of the chapter as well as in the exercises. Horizonal asymptotes, limits at infinity, vertical asymptotes and infinite limits are discussed.
   1. The limit of a function (56)
   2. Theorems on limits of functions (64)
   3. One side limits (73)
   4. Infinite limits (78)
   5. Limits at infinity (88)
   6. Continuity of a function at a number (98)
   7. Continuity of a composite function and continuity on an interval (107)
   8. Continuity of the trigonometric functions and the squeeze theorem (114)
   9. Proofs of some theorems on limits of functions (Supplementary) (122)
   10. Additional theorems on limits of functions (131)

   Continue reading “Calculus Outline – Louis M. Leithold 6/e”

Found my outline! It’s the table of contents for the textbook The Calculus with Analytic Geometry, 6th Edition, by Louis M. Leithold. This was the text from which I learned everything of undergraduate calculus, from limits to Green’s functions and basis vectors and multiple integrals, well over a decade ago. My first copy had eventually become all dogeared and coffee-stained from hours of poring through the text over three semesters of coursework; the second copy fared no better, as it saw me through the last of three semesters, and one more taking in differential equations.  I’d typed in the table of contents to serve as my weekly reading guide; those were busy days.

What’s particularly good about TCWAG is that it provides clear step-by-step proofs of all theorems and major results, unlike free texts offered up by MIT through it’s Open Courseware program (see Calculus, by Gilbert Strang, at http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/). Only rarely did we encounter use of results that could not be developed as part of an introductory calculus course – these were clearly identified as, for example, in the case of the completeness axiom, which could only be fully developed in a course in real analysis.

In all there are 18 chapters, which we’d covered at a rate of about six chapters per semester. This outline will serve as my guide to retrieving the calculus in the next few months.

1. FUNCTIONS OF A SINGLE REAL VARIABLE– Limits and continuity – Complete discussion includes proofs of all theorems in the body of the chapter as well as in the exercises. Horizonal asymptotes, limits at infinity, vertical asymptotes and infinite limits are discussed.
   1. The limit of a function (56)
   2. Theorems on limits of functions (64)
   3. One side limits (73)
   4. Infinite limits (78)
   5. Limits at infinity (88)
   6. Continuity of a function at a number (98)
   7. Continuity of a composite function and continuity on an interval (107)
   8. Continuity of the trigonometric functions and the squeeze theorem (114)
   9. Proofs of some theorems on limits of functions (Supplementary) (122)
   10. Additional theorems on limits of functions (131)
2. The derivative and differentiation– The classical geometrical interpretation of the derivative, detailed derivative evaluation, concise proofs of theorems on differentiation, derivatives of trig functions, and the Chain Rule.
   1. The tangent line and the derivative (139)
   2. Differentiability and continuity (148)
   3. Theorems on differentiation of algebraic functions (156)
   4. Rectilinear motion and the derivative as a rate of change (163)
   5. Derivatives of the trigonometric functions (173)
   6. The derivative of a composite function and the Chain Rule (181)
   7. The derivative of the power function for rational exponents (190)
   8. Implicit differentiation (195)
   9. Related rates (199)
   10. Derivatives of higher order (205)
3. Function extremaTechniques of graphing The differential – Theorems on extrema and graph behavior motivate a complete method for function graph sketching. Differentials are treated in this section, adjacent to differentiation subject matter.
   1. Maximum and minimum function values (217)
   2. Applications involving an absolute extremum on a closed interval (224)
   3. Rolle’s theorem and the Mean-value theorem (230)
   4. Increasing and decreasing functions and the first-derivative test (236)
   5. Concavity and points of inflection (241)
   6. The second-derivative test for relative extrema (249)
   7. Drawing the sketch of the graph of a function (254)
   8. Further treatment of absolute extrema and applications (260)
   9. The differential (269)
   10. Numerical solutions Newton’s method (277)
4. The definite integralIntegration – Indefinite integration, area, the definite integral, the Fundamental Theorem of the Calculus. Trapezoidal and parabolic rules, formulas for error bound estimation.
   1. Antidifferentiation (286)
   2. Some techniques of antidifferentiation (295)
   3. Differential equations and rectilinear motion (303)
   4. Area (312)
   5. The definite integral (324)
   6. Properties of the definite integral (331)
   7. The mean-value theorem for integrals (340)
   8. The fundamental theorems of the calculus (344)
   9. Area of a plane region (352)
   10. Numerical integration (supplementary) (359)
5. Applications of the definite integral– Evaluation techniques and principles are discussed, supported by concise motivation and explanations. Applications in physics.
   1. Volumes by slicing, disks, and washers (374)
   2. Cylindrical-shell method (383)
   3. Length of arc of the graph of a function (388)
   4. Center of mass of a rod (394)
   5. Centroid of a plane region (400)
   6. Work (407)
   7. Liquid pressure (supplementary) (413)
6. Inverse, logarithmic and exponential functions– Includes a concise definition of irrational numbers. Applications.
   1. Inverse functions (422)
   2. Inverse function theorems Derivative of a function inverse (431)
   3. The natural logarithmic function (439)
   4. Logarithmic differentiation Integrals yielding ln(x) (449)
   5. The natural exponential function (455)
   6. Other exponential and logarithmic functions (463)
   7. Applications (469)
   8. First-order linear differential equations (supplementary) (481)
   9. Review (492)
7. Inverse trigonometric and hyperbolic functions.
   1. The inverse trigonometric functions (496)
   2. Derivatives of the inverse trigonometric functions (503)
   3. Integrals yielding inverse trigonometric functions (510)
   4. Hyperbolic functions (514)
   5. Inverse hyperbolic functions (supplementary) (523)
   6. Review (527)
8. Techniques of integration– Computational methods encountered in practical problems. Crucial examples illustrate the principles involved.
   1. Integration by parts (531)
   2. Of powers of sine and cosine (537)
   3. Of powers of tangent, cotangent, secant and cosecant (542)
   4. By trigonometric substitution (545)
   5. By partial fractions: linear (551)
   6. By partial fractions: quadratic denom. fact (561)
   7. Miscellaneous substitutions (566)
   8. Integrals yielding inverse hyperbolic functions (supplementary) (570)
   9. Review (575)
9. The conic sections and polar coordinates.
   1. The parabola and translation of axes (578)
   2. The ellipse (586)
   3. The hyperbola (594)
   4. Rotation of axes (604)
   5. Polar coordinates (608)
   6. Graphs of equations in polar coordinates (614)
   7. Area of a region in polar coordinates (625)
   8. A unified treatment of conic sections and polar equations of conics (629)
   9. Tangent lines of polar curves (supplementary) (638)
   10. Review (647)
10. Indeterminate forms, improper integrals, and Taylor’s formula– Concepts supporting a discussion on infinite series are assayed. Probability density function.
    1. The indeterminate form 0/0 (651)
    2. Other indeterminate forms (660)
    3. Improper integrals with infinite limits of integration (665)
    4. Other improper integrals (673)
    5. Taylor’s formula (677)
    6. Review (684)
11. INFINITE SERIES– Sequences and infinite series – Theorems. Tests for convergence of a series.
    1. Sequences (687)
    2. Monotonic and bounded sequences (694)
    3. Infinite series of constant terms (700)
    4. Infinite series: Four theorems (709)
    5. Infinite series of positive terms (713)
    6. The integral test (723)
    7. Alternating series (726)
    8. Absolute and conditional convergence Ratio test Root test (731)
    9. Summary of convergence tests for infinite series (738)
    10. Review (740)
12. Power series.
    1. Introduction to power series (743)
    2. Differentiation of power series (750)
    3. Integration of power series (760)
    4. Taylor series (767)
    5. The binomial series (776)
    6. Review (780)
13. VECTOR-VALUED AND MULTIVARIABLE FUNCTIONS– Vectors in the plane and parametric equations.
    1. Vectors in the plane (783)
    2. Dot product (794)
    3. Vector-valued functions Parametric equations (801)
    4. Calculus of vector-valued functions (808)
    5. Length of arc (814)
    6. The unit tangent and normal vectors Arc length as parameter (820)
    7. Curvature (824)
    8. Plane motion (832)
    9. Tangential and normal components of acceleration (838)
    10. Review (842)
14. 3D Vectors and solid analytic geometry.
    1. R^3 space (846)
    2. Vectors in R^3 (852)
    3. Planes (861)
    4. Lines in R^3 (868)
    5. Cross product (873)
    6. Cylinders and surfaces of revolution (883)
    7. Quadric surfaces (888)
    8. Curves in R^3 (894)
    9. Cylindrical and spherical coordinates (901)
    10. Review (905)
15. INTRODUCTION TO MULTIVARIATE CALCULUS– Differential calculus of functions of more than one variable – Extensions of 1V calculus.
    1. Functions of more than one variable (908)
    2. Limits (917)
    3. Continuity (927)
    4. Partial derivatives (931)
    5. Differentiability Total differentials (939)
    6. The Chain Rule (949)
    7. Higher-order partial derivatives (956)
    8. Sufficient conditions for differentiability (963)
    9. Review (968)
16. Directional derivatives, gradients, and applications of partial derivatives– Vector fields. Solution of extrema problems and Lagrange multipliers. Exact differential equation solution.
    1. Directional derivatives and gradients (972)
    2. Tangent planes and normals to surfaces (979)
    3. Extrema of functions of two variables (983)
    4. Lagrange multipliers (997)
    5. Obtaining a function from its gradient and exact differentials (1003)
    6. Review (1011)
17. Multiple integration
    1. The double integral (1014)
    2. Double and iterated integrals (1019)
    3. Center of mass and moments of inertia (1026)
    4. The double integral in polar coordinates (1031)
    5. Area of a surface (1036)
    6. Triple integrals (1041)
    7. Triple integrals in cylindrical and spherical coordinates (1046)
    8. Review (1052)
18. Introduction to the calculus of vector fields– Intuitive appproach to problems in physics and engineering.
    1. Vector fields (1056)
    2. Line integrals (1064)
    3. Line integrals independent of path (1072)
    4. Green’s theorem (1082)
    5. Surface integrals (1095)
    6. Gauss’s Divergence Theorem and Stokes’s Theorem (1102)
    7. Review (1108)

I’ll head off to a local bookstore in about thirty minutes to see whether this text is even still in distribution – I’m sure that I’ll at least be able to order a copy from overseas.

 

Flex (http://flex.sourceforge.net/manual/Patterns.html#Patterns) for parsing expressions. Algebra and trigonometry again.

Can’t sleep: My mind is wandering about in my skull, a kitty among the bins. I can’t make up my mind how to represent an atomic algebraic expression term.

Clearly, an expression term, and an elementary binary operator are two different kinds of objects. What do I make, then, of an expression term such as sin(x)? It’s a function operator taking a single argument. I’d like to be able to manipulate a pair of rational expressions like

 

How about 2x? It’s an elementary product expression consisting of a constant and a variable. I’m beginning to think each subexpression ought to be represented as an object, now – so that I could replace the factor 2 with an expression, for example. This pretty much puts paid to representing an algebraic polynomial term as a struct:

struct Monomial {
  public:
    integer coeff;
    Variable x;
    Exponent t;
}

Each member coeff, x,  and t would then be an Expr:

struct Monomial {
  protected:
    Expr coeff;
    Expr x;
    Expr t;
}

Then I should represent elementary operations as functors as well, perhaps. Hmm. This eliminates the need for a Monomial type:

Expr x('x'), X;
X = 2 * x ^ 3;
cout << X.strict().toString() << endl; // 2*(x^3)
cout << X.toHumanString() << endl; // 2x3
cout << X.toMathML() << endl; // Something like 2x(super)3

I need to look into operator associativity rules and find out whether any combination of member and nonmember functions will cause type coercion of the RHS into an Expr type.

***

My goal is to be able to create simple code to generate elementary algebra expressions programmatically, like so:

Expr a('a', '1/(x+2)');
Expr b('b', '2/(x+7)');
Expr c;
c = a * b;
cout << c.toMathML() << endl;
cout << c.execute().toMathML() << endl; // 1/(x^2 + 9x + 14)

Then a simple, short chunk of code could do and generate an online algebra workbook of arbitrary complexity:

string generate_rational_fraction_sum( int terms, int glbpower, int lubpower ) {
  Expr rfs;
  Expr numerator;
  Expr denominator;
  Expr monomial('x');
  for ( int i = 0 ; i < terms ; i++ ) {
    integer r = rand(glbpower, lubpower);
    integer q = rand(1,10);
    denominator.clear();
    for ( int j = 0 ; j < r ; j++ ) {
      // Generate an order-j polynomial expression for the denominator
      integer k = rand(1,10);
      denominator += k * monomial ^ j;
    }
    numerator = rand(1,10);
    rfs += numerator / denominator;
  }
  return rfs.execute().toMathML();
}

Trigonometric identities – gone.  Techniques for polynomial factoring – gone. Ditto for logarithmic and exponential functions. Drat!

I guess recovering the calculus is basically going to take a wholesale rebootstrap of my entire maths education. Granted, a lot of it is going to be recovery rather than reintroduction. The skills have atrophied, and the surety of technique gone, but the memory of how to construct a proof, for example – the concept of inductive logic – isn’t completely lost to me.

This feels like having fallen off a mountain, nevertheless.

“Anybody have pitons?” I got pitons. The idea I have is to do both relearn algebra and trigonometry and write the software tools to help me do the math exercises. Instead of just the old pen and paper, I’d like to make use of a LaTeX engine and parser to enable me to solve these problems like I’d do on paper, but onscreen instead. I think of it basically as a way to generate problem sets and grade them on the fly to get feedback quicker (and to offset the tedium of interacting with the computer using an interface – a keyboard and maybe the mouse – that doesn’t allow easy math symbol input).

Much as I’d like to reprise Vance in its’ entirety, I’ll need to use a University course outline to guide the structure of my application. There were bits of Vance that led into complex analysis, real analysis, and linear algebra that I’d like to include as part of my course flow. I’m imagining an interactive Vance that does for algebra and trigonometry what Push Pop Press did for Al Gore’s book on the environment, and digital media should be able to be molded to the purpose: Javascript and HTML5 have evolved quite a bit since 1996, and might just enable a Web interactive application to do just  that.

Can’t wait to start on elementary physics, a la Angry Birds.