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Decimal View of Real Analysis
Have a good look at the above table. Then note most of this site area has been duplicated and posted online in the new Calculus Introduction site area. The Appendices -- Beyond Calculus
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| In Chapter 15, a short section
Limits & Algebra explains or describes the algebraic
evaluation of limits. This section can be read out of sequence and
should be to develop algebraic thinking skills. Repetitive examples in this section lead to a formula for the slope or derivative of the quadratic function y = x2 while introducing and illustrating the algebraic way of writing and reasoning vital to understanding the limit definition of slopes and derivatives. The repetition leads a student to an algebraic pattern and to another aspect or facet algebraic way of writing and reasoning. |
Chapter 16 What is Velocity (poster material) provides motivation for the standard approximation for the velocity non-constant speed motion, and then says if the approximation converges to a limit, that limit is the velocity (a slope or derivative). That is a second twist of the typical kind in calculus or its interpretations. Saying how to compute or approximate a quantity, here the velocity in the limit, defines it if there is convergence.
Chapter 17 What is Area (poster material) provides motivation for the standard approximation for the area of a region and then says if the approximation converges to a limit, that limit is the area. That is a third twist of the typical kind in calculus or its interpretations. Saying how to compute or approximate a quantity, here the area in the limit, defines it if there is convergence. Area here is approximated by covering with squares.
The first section of Chapter 18 Integration (poster material) moves from approximating areas under a curve y = f(x) > 0, a special region, with squares to a covering with rectangles and thus introduces the Riemann Sum approximation. The second second shows or suggest how to compute the limit, assuming it exists through anti-differentiation, that is a reversal of the slope computation process. The anti-differentiation process is justified in some cases by the Second fundamental theorem of calculus. The assumption that the limit of area approximations exists is justified by the first fundamental theorem of calculus. The proof of the latter is given in the last appendix as an appetizer for advanced students.
The middle section of Chapter 18 introduces a function F(x) by defining it as the area under a curve y = f(x) between two points a and x. The last section of chapter 18 describes five more ways to define functions, ways often met in mathematics before, in and after calculus.
Chapter 19 Logs & Powers introduces or defines the natural logarithm, another function, as the (signed) area under a curve. This definition is similar to that which appears in the middle section of Chapter 18, the discussion or derivation of the second fundamental theorem of calculus. Further logarithms are defined (briefly, too briefly) in terms of the natural logarithm, while the exponential function and one non-negative real number raised to a power (another real number) are introduced ro defined with the aid of the inverse to the natural logarithm.
Chapter 20 What's Next describes the next chapters.
Chapter 21 Add Vectors shows how to add vectors in the plane, and attempts to provide motivation for it. The presentation here is an exploration of ideas which I wanted to clarify for myself, if not the reading.
Chapter 22 Complex Numbers (Basic Ideas) introduces this numbers geometrically.
Elements of this chapter appear in the simplified treatment of complex numbers (& trig), posted online at this site after the completion of Volume 3.
Chapter 23 Complex Numbers (Links to Trig) first points to the well-known simplification of trigonometry given by the use of complex number function cis(A) = cos(A) + i sin(A) = exp(iA) in higher mathematics and the mathematical disciplines, engineering and physics included.
The last part of Chapter 23 Complex Numbers (Links to Trig) explores different ways to establish the complex number, distributive law for multiplication over addition. The simplified treatment of complex numbers (& trig), posted online at this site after the completion of this book, Volume 3 is simpler. But I am still puzzled regarded the optimal way to develop complex numbers and trigonometry impurely from a mix of assumptions about arithmetic and geometry. In pure mathematics, trigonometric functions may be defined without reference or dependence on geometric diagrams. However, novices need diagrams of one kind or another for their first comprehension of trig functions.
Chapter 24 Complex Logs etc states formulas for logarithms, exponentials and powers of complex numbers, and formulas for the hyperbolic functions. It provides no more information about the functions.
www.whyslopes.com
Real Analysis - Decimal View
Here are the Appendices from Volume 3, Why Slopes and More Math, Chapters 14 to 19 in Vol 3 are related. Here is a reference for college or university mathematics, electrical engineering and physics.
A. What's Next B. Pigeon Hole Principle B. Bolzano-Weierstrass C1. Triangle Inequality C2. Triangle Inequality C. More T.Inequality D. Sets & Sequences D. Monotone Sequences E. Limits, Properties E Limits & Error Control F. Continuous Functions F. Closed Range Thm F. Intermediate Val. Thm F. Compactness Thm F. Equicontinuity Thm F Extreme Value Thm G. Rolle's Theorem etc G. Mean Val. Thm. G. Constant Difference Thm G. Lipschitz Continuity I PS: One Sided Range Theorems G. Velocity Revisited G. Sufficient Conditions H. Riemann Sums Conv H. Lipschitz Continuity II The site area More Calculus contains a one-sided theorem with proof that should be of interest too.
Vol 1A Logic Postscripts
online only:-Proof by Absurdity alias proof by contradiction
How the demand for consistency supports the law of the excluded middle
Reality versus or with the aid of Imagination
Science, Engineering & Math Students: Have you seen a simpler geometric introduction to complex numbers? ( java applet included) . Can you explain what is a variable without using a symbol? Can you derive trig expression for dot & cross & cosine law from complex number properties? For truth tables and indirect methods of reason, see chapters 19-24 & postscripts in Pattern Based Reason and visit Volume 1A, Pattern Based Reason, striving for objectivity, the empirical challenge & limits.
Vol 1A Postscripts
online onlyProof by Absurdity alias proof by contradiction
How the demand for consistency supports the law of the excluded middle
Help Me Learn/Teach;
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