# Handbook of Mathematical Formulas

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Handbook of Mathematical Formulas presents a compilation of formulas to provide the necessary educational aid. This book covers the whole field from the basic rules of arithmetic, via analytic geometry and infinitesimal calculus through to Fourier's series and the basics of probability calculus. Organized into 12 chapters, this book begins with an overview of the fundamental notions of set theory. This text then explains linear expression wherein the variables are only multiplied by constants and added to constants or expressions of the same kind. Other chapters consider a variety of topics, including matrices, statistics, linear optimization, Boolean algebra, and Laplace's transforms. This book discusses as well the various systems of coordinates in analytical geometry. The final chapter deals with algebra of logic and its development into a two-value Boolean algebra as switching algebra. This book is intended to be suitable for students of technical schools, colleges, and universities.

## Table of Contents

0. Mathematical Signs and Symbols

0.1. Mathematical Signs

0.2. Symbols Used in the Theory of Sets

0.3. Symbols of Logic

1. Arithmetic

1.1. Set Theory

1.1.1. Fundamental Notions

1.1.2. Set Operations

1.1.3. Mappings, Cardinality

1.2. Real Numbers

1.2.1. General

1.2.2. Irrational Numbers

1.2.3. Binomial Coefficients, Binomial Theorem

1.3. Imaginary or Complex Numbers

1.3.1. Imaginary Numbers

1.3.2. Complex Numbers in Arithmetical Form

1.3.3. Complex Numbers in a Goniometric Form

1.3.4. Complex Numbers in Exponential Form

1.3.5. Natural Logarithms of Complex and Negative Numbers

1.3.6. Graphical Methods

1.4. Proportions

1.5. Logarithms

1.5.1. General

1.5.2. Rules for Calculating with Logarithms

1.5.3. The Use of Logarithm Tables for Finding Common Logarithms

1.6. Combinatoric Analysis

1.6.1. Permutations

1.6.2. Variations

1.6.3. Combinations

1.7. Per Cent Calculation, Interest Calculation

1.7.1. Per Cent (Per Mille) Calculation

1.7.2. Interest Calculation

1.8. Sequences and Series

1.8.1. General

1.8.2. Arithmetic Sequences and Series

1.8.3. Geometric Sequences and Series

1.8.4. Compound Interest Calculation

1.8.5. Annuities

1.9. Determinants

1.9.1. General

1.9.2. Theorems on Determinants

1.9.3. Applications of Determinants

1.10. Matrices

1.10.1. General

1.10.2. Theorems on Matrices

1.10.3. Applications

2. Equations, Functions, Vectors

2.1. Equations

2.1.1. General

2.1.2. Algebraic Equations in One Variable

2.1.3. Transcendental Equations

2.1.4. Approximation Methods for Determining the Roots of an Equation

2.1.5. Systems of Equations

2.2. Inequalities

2.3. Functions

2.3.1. General

2.3.2. Further Methods of Analytic Representation

2.3.3. Graphical Representation of Functions

2.4. Vector Calculus

2.4.1. General

2.4.2. Multiplication of Vectors

2.4.3. Geometrical Applications of Vector Calculus

2.5. Reflection in a Circle, Inversion

3. Geometry

3.1. General

3.2. Planimetry

3.2.1. Triangle ABC

3.2.2. Quadrilaterals

3.2.3. Polygons (n-Sided Polygons)

3.2.4. Circle

3.3. Stereometry

3.3.1. General Theorems

3.3.2. Solids Bounded by Plane Surfaces

3.3.3. Solids Bounded by Curved Surfaces

3.4. Goniometry, Plane Trigonometry, Hyperbolic Functions

3.4.1. Goniometry

3.4.2. Trigonometric Formulas for Oblique-Angled Triangles

3.4.3. Goniometric Equations

3.4.4. Inverse Trigonometric Functions

3.4.5. Hyperbolic Functions

3.4.6. Inverse Hyperbolic Functions

3.5. Spherical Trigonometry

3.5.1. General

3.5.2. Right Spherical Triangle

3.5.3. Oblique Spherical Triangle

3.5.4. Mathematical Geography

4. Analytical Geometry

4.1. Analytical Geometry of the Plane

4.1.1. The Various Systems of Coordinates

4.1.2. Points and Line Segments

4.1.3. Straight Line

4.1.4. Circle

4.1.5. Parabola

4.1.6. Ellipse

4.1.7. Hyperbola

4.1.8. The General Equation of the Second Degree in x and y

4.2. Analytical Geometry of Space

4.2.1. The Various Systems of Coordinate

4.2.2. Points and Line Segments in Space

4.2.3. Planes in Space

4.2.4. Straight Lines in Space

4.2.5. Surfaces of the Second Order

4.2.6. The General Equation of the Second Degree in x, y and z

5. Differential Calculus

5.1. Limits

5.2. Difference Quotient, Differential Quotient, Differential

5.3. Rules for Differentiation

5.4. Derivatives of the Elementary Functions

5.5. Differentiation of a Vector Function

5.6. Graphical Differentiation

5.7. Extrema of Functions (Maxima and Minima)

5.8. Mean-Value Theorems

5.9. Indeterminate Expressions

6. Differential Geometry

6.1. Plane Curves

6.1.1. Main Elements of Plane Curves

6.1.2. A Few Important Plane Curves

6.2. Space Curves

6.3. Curved Surfaces

7. Integral Calculus

7.1. Definition of the Indefinite Integral

7.2. Basic Integrals

7.3. Rules of Integration

7.4. A Few Special Integrals

7.4.1. Integrals of Rational Functions

7.4.2. Integrals of Irrational Functions

7.4.3. Integrals of Trigonometric Functions

7.4.4. Integrals of the Hyperbolic Functions

7.4.5. Integrals of Exponential Functions

7.4.6. Integrals of the Logarithmic Functions

7.4.7. Integrals of the Inverse Trigonometric Functions (Arc Functions)

7.4.8. Integrals of the Inverse Hyperbolic Functions (Area Functions)

7.5. Definite Integral

7.5.1. General

7.5.2. Mean-Value Theorems for Integral Calculus

7.5.3. Geometrical Interpretation of the Definite Integral

7.5.4. Methods of Approximation for Definite Integrals

7.5.5. Graphical Integration

7.5.6. Improper Integrals

7.5.7. A Few Definite Integrals

7.5.8. Applications of the Definite Integral

7.6. Line Integral

7.6.1. Line Integrals in the Plane

7.6.2. Line Integrals in Space

7.6.3. Line Integral of a Vector

7.7. Multiple Integrals

7.7.1. Double Integrals

7.7.2. Triple Integrals

8. Differential Equations

8.1. General

8.2. Ordinary Differential Equations of the First Order

8.2.1. Separation of Variables

8.2.2. Homogeneous Differential Equations of the First Order

8.2.3. Inhomogeneous Differential Equations of the First Order

8.2.4. Total (Exact) Differential Equations of the First Order

8.2.5. Integrating Factors

8.2.6. Bernoulli Differential Equation

8.2.7. Clairaut Differential Equation

8.2.8. Riccati Differential Equation

8.3. Ordinary Differential Equations of the Second Order

8.3.1. Special Cases

8.3.2. Linear Homogeneous Differential Equation of the Second Order with Constant Coefficients

8.3.3. Linear Homogeneous Differential Equation of the Second Order with Variable Coefficients

8.3.4. Euler Differential Equation

8.3.5. Linear Inhomogeneous Differential Equation of the Second Order with Constant Coefficients

8.3.6. Linear Inhomogeneous Differential Equation of the Second Order with Variable Coefficients

8.4. Ordinary Differential Equations of the Third Order

8.4.1. Linear Homogeneous Differential Equation of the Third Order with Constant Coefficients

8.4.2. Linear Inhomogeneous Differential Equation of the Third Order with Constant Coefficients

8.5. Integration of Differential Equations by Power Series

8.6. Partial Differential Equations

8.6.1. Simple Partial Differential Equations

8.6.2. Linear Partial Differential Equation of the First Order for z = f(x, y)

9. Infinite Series, Fourier Series, Fourier Integral, Laplace Transforms

9.1. Infinite Series

9.1.1. General

9.1.2. Convergence Criteria

9.1.3. Some Infinite Convergent Series

9.1.4. Power Series

9.1.5. Approximation Formulas

9.2. General Statements on Fourier Series, Fourier Integrals, and Laplace Transforms

9.3. Fourier Series

9.4. Fourier Integral, Example of Calculation

9.5. Laplace Transforms

9.6. Exployment of Laplace Transforms; Solution of Differential Equations

9.7. Table of Correspondences of Some Rational Laplace Integrals

10. Theory of Probability; Statistics; Error Calculation; Mathematical Analysis of Observations

10.1. Theory of Probability

10.2. Statistics

10.3. Error Calculations

10.4. Calculus of Observations

11. Linear Optimization

11.1. General

11.2. Graphical Procedure

11.3. Simplex Procedure (Simplex Algorithm)

11.4. Simplex Table

12. Algebra of Logic (Boolean Algebra)

12.1. General

12.2. Arithmetical Laws, Arithmetical Rules

12.3. Further Possibilities of Interconnecting Two Input Variables (Lexigraphic Order)

12.4. Normal Forms

12.5. Karnaugh Tables

Appendix: The Dual System (Dyadic System)

The Roman Decimal System

Greek Alphabet

Frequently Used Numbers and their Common Logarithms

Index

## Product details

- No. of pages: 528
- Language: English
- Copyright: © Academic Press 1974
- Published: November 28, 1974
- Imprint: Academic Press
- eBook ISBN: 9781483267425

## About the Author

### Hans-Jochen Bartsch

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