Calculus for the Systems Architect
An Essay on the Eternal
1st edition, by James A. Green, Greenwood Research.
[ ] Large Hardback, ISBN-13: 978-1-890121-22-8 (ISBN 1-890121-22-3), 59.00 dollars.
[ ] Large Paperback, ISBN-13: 978-1-890121-98-3 (ISBN 1-890121-98-3), 52.00 dollars.
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I. Review of Trigonometry.
__1.1 The Euclidean Foundation.
__1.2 The Theorem of Pythagoras.
__1.3 Similar Angles and Parallel Lines
__1.4 The Ratio Pi and It's Value
__1.5 The Trigonometric Functions and Their Inverses
__1.6 The Limits of the Trigonometric Functions
__1.7 The Function Concept and More Complex Trigonometric Functions
__1.8 The Trigonometric Identities
II. Analytical Geometry and Vectors
__2.1 Systems of Coordinates - Historical Reflections
__2.2 The Circle
__2.3 The Ellipse
__2.4 The Line
__2.5 The Parabola and Hyperbolas
__2.7 Polynomial Functions
__2.8 Systems of Coordinates
III. Differentiation of Polynomials
__3.1 The Slope of the Tangent to a Curve
__3.2 Orders of Differentiation
__3.3 Differentiation of a Line
__3.4 Differentiation of Powers of X
__3.5 Linear Properties of Differential Operators
IV. Rules of Differentiation
__4.1 Euler's Constant e
__4.2 Natural Logarithms
__4.3 Logarithms to Other Bases
V. Rules of Differentiation
__5.1 Fundamental Rules of Differentiation
__5.2 The Product Rule for Functions
__5.3 The Rule for Quotients of Functions
__5.4 The Chain Rule for Nested Functions
__5.5 Differentiation of Natural Logarithms
__5.6 Differentiation of Numbers to the Xth Power
VI. Differentiation of Trigonometric Functions
__6.1 Differentiation of cos(a).
__6.2 Differentiation of sin(a).
__6.3 Differentiation of tan(a).
__6.4 Differentiation of sec(a) and cosec(a).
__6.5 Inverse Functions
__7.1 The Area Under a Curve
__7.2 The Fundamental Theorem of the Calculus
__7.3 Rules of Integration
__7.4 Trigonometric Integrals
__7.5 Integrals with Infinite Bounds
VIII. Techniques of Integration
__8.2 Change of Variables and Substitution
__8.3 Trigonometric Substitution
__8.4 Integration by Parts
IX. Maclaurin and Taylor Series
__9.1 Introduction to Series Expansions
__9.2 Maclaurin Series
__9.3 Important Examples of Maclaurin's Series
__9.4 Taylor's Series
__9.5 Remainder Theorem for Taylor's Series
X. Applications of Calculus
__10.1 Volumes & Masses of Solids: The Stars Above
__10.2 Volumes and Surfaces of Revolution
__10.3 Moments of Inertia
__10.4 Elementary Differential Equations
__10.5 Equations of Motion
About the Author
Appendix A. Archimedes Exterior Sums
Appendix B. Limits of Trigonometric Functions
Appendix C. Measuring the Earth's Radius in Spain
Appendix D. Geometric Proof of Trig Identities
Appendix E. Natural Constants and Measures
Appendix F. Tables of Calculus Definitions and Results
___Areas and Volumes
_____The Definite Integral
_____Exponential and Logarithmic Integrals
_____Integrals of Trigonometric Functions
_____Integrals including [a2 - u2]1/2
_____Integrals of Inverse Trigonometric Functions
_____Integrals Containing Hyperbolic Functions
_____Integrals Including [u2 +- a2]1/2
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List of Illustrations
1.1 Inscribed Square for the Theorem of Pythagoras
1.2 Parallel Lines Cut by a Ray
1.3 Equal Angles from Parallels Cut by a Ray
1.4 Measuring the Radius of the Earth
1.5 Columbus' Method for Tricking Queen Isabella
1.6 Pi as the Ratio of Circumference to Diameter
1.7 Chinese Method for Pi as the Area of a Circle
1.8 Chinese Method for Computing Pi.
1.9 Exterior Upper Sun and Interior Lower Sum for
____Computing the value of Pi between limits.
1.10 Pi by the Method of Archimedes: Interior Sum
1.11 Obtaining Archimedes Trig Tables
1.12 Trigonometric Functions in the 1st Quadrant
1.13 Trigonometric Functions in the 2nd, 3rd, and 4th Quadrants
1.14 Plot of sin(a) and cos(a)
1.15 The Tangent Function tan(a)
2.1 The Circle of Radius R
2.2 The Circle of Radius R at P
2.3 The Ellipse of Semi-Major Axis A and Semi-Minor Axis B
2.4 Focal Representation of an Ellipse
2.5 Computing the Semi-Major Axis B
2.6 The Equation of a Line and Its Slope
2.7 Parabola and Surrounding Hyperbolas
2.8 Decomposition of a Vector V
2.9 The Scaler Field T(r) inside a Star
2.10 the Vector Field Associated with an Electric Field
2.11 The Vector Dot Product
2.12 The Vector Cross-Product
2.13 Cartesian Coordinates
2.14 Polar Coordinates and Cartesian Coordinates
2.15 Spherical Coordinates
2.16 Cylindrical Coordinates
2.17 Hyperbolic Coordinates
3.1 The Slope of a Line Tangent to a Curve
3.2 The Derivative of a Line
3.3 Differentiation of the Square of X.
6.1 Angles in the cos(a + da) - cos(a) Problem
6.2 da[sin(a)] = cos(a) - cos(a + da)
6.3 Inverse Functions
7.1 Lower and Upper Sums for the Square of X.
10.1 Surface of Revolution Radius Function
10.2 Associated Volume of Revolution
10.3 Revolving Wheel of Constant Density
10.4 Sharp Revolving Wheel with Triangular Cross-Section
10.5 Build-Up and Decay of Motion in a Torque-Motor Driven Wheel
A.1 The Exterior Polygon for Archimedes Upper Sums
B.1 Trigonometric Functions
C.1 Measurement of the Earth's Radius Well Away from the Equator
D.1 Geometric Construction for Proving Trig Identities
List of Tables
1. Basic Trig Identities
2. Table of Ellipse Identities
3. Table of Exponential and Logarithmic Identities
4. Fundamental Rules of Differentiation
5. Differentiation of Trigonometric Identities
6. Properties of Trigonometric Integrals
1. Orion Standing in the C/pi Winter Crescent.
2. Portrait of the Author with an apparatus for measuring the distance to the Sun.
3. Centaurus gesturing over the Balance of Libra.
4. Portrait of the author, study desk.
5. Portrait of Christopher Columbus.
6. Constellation Lyre as Pi-Symbol.
7. Cygnus waves above Lyre-Pi as the C/pi Winter Crescent Rises in the East.
8. Orion standing in the C/pi Winter Crescent.
9. Portrait of Archimedes, Oriental Manuscripts on Pi.
10. Integral Signs as Violin Holes.
11. The author as a systems engineer in Florida.
12. Cygnus, Lyre-Pi, and Hercules in the Sight of Aquila.
13. "Pi-sees" about to lasso Aquarius.
14. The Great Seal of the United States and the Rise of Capella.
15. "Heart-of-the-Matter" star cluster and Lyre-Pi.
16. James Clerk Maxwell's Visionary Symbolist Creator
17. Shin bone of a Wolf as 1st Tally Stick & Pi to 2000 places.
18. Areas and Volumes
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