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Guide to Essential Math A review for Physics, Chemistry and Engineering Students written by
S. M. Blinder , University of Michigan, Ann Arbor, USA.
This book reminds students in junior, senior and graduate level courses in physics, chemistry and engineering of the math they may have forgotten (or learned imperfectly) which is needed to succeed in science courses. The focus is on math actually used in physics, chemistry and engineering, and the approach to mathematics begins with 12 examples of increasing complexity, designed to hone the student's ability to think in mathematical terms and to apply quantitative methods to scientific problems. By the author's design, no problems are included in the text, to allow the students to focus on their science course assignments. - Highly accessible presentation of fundamental mathematical techniques needed in science and engineering courses - Use of proven pedagogical techniques develolped during the author's 40 years of teaching experience - illustrations and links to reference material on World-Wide-Web - Coverage of fairly advanced topics, including vector and matrix algebra, partial differential equations, special functions and complex variables.
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Guide to Essential Math A review for Physics, Chemistry and Engineering Students written by
S. M. Blinder
cover the following topics.
1. Mathematical Thinking
1.1 The NCAA March Madness Problem
1.2 Gauss and the Arithmetic Series
1.3 The Pythagorean Theorem
1.4 Torus Area and Volume
1.5 Einstein’s Velocity Addition Law
1.6 The Birthday Problem
1.7 Fibonacci Numbers and the Golden Ratio
1.8 vp in the Gaussian Integral
1.9 Function Equal to Its Derivative
1.10 Log of N Factorial for Large N
1.11 Potential and Kinetic Energies.
1.12 Riemann Zeta Function and Prime Numbers
1.13 How to Solve It
1.14 A Note on Mathematical Rigor
2. Numbers
2.1 Integers.
2.2 Primes.
2.3 Divisibility
2.4 Rational Numbers
2.5 Exponential Notation.
2.6 Powers of 10
2.7 Binary Number System.
2.8 Infinity
3. Algebra
3.1 Symbolic Variables
3.2 Legal and Illegal Algebraic Manipulations.
3.3 Factor-Label Method
3.4 Powers and Roots.
3.5 Logarithms.
3.6 The Quadratic Formula
3.7 Imagining
3.8 Factorials, Permutations, and Combinations
3.9 The Binomial Theorem.
3.10 e Is for Euler
4. Trigonometry
4.1 What Use Is Trigonometry?.
4.2 The Pythagorean Theorem.
4.3 p in the Sky
4.4 Sine and Cosine
4.5 Tangent and Secant
4.6 Trigonometry in the Complex Plane.
4.7 de Moivre’s Theorem
4.8 Euler’s Theorem
4.9 Hyperbolic Functions
5. Analytic Geometry
5.1 Functions and Graphs
5.2 Linear Functions.
5.3 Conic Sections
5.4 Conic Sections in Polar Coordinates.
6. Calculus
6.1 A Little Road Trip
6.2 A Speedboat Ride.
6.3 Differential and Integral Calculus
6.4 Basic Formulas of Differential Calculus
6.5 More on Derivatives
6.6 Indefinite Integrals
6.7 Techniques of Integration
6.8 Curvature, Maxima, and Minima
6.9 The Gamma Function
6.10 Gaussian and Error Functions
7. Series and Integrals
7.1 Some Elementary Series
7.2 Power Series
7.3 Convergence of Series
7.4 Taylor Series
7.5 L’Hopital’s Rule ˆ
7.6 Fourier Series
7.7 Dirac Deltafunction
7.8 Fourier Integrals
7.9 Generalized Fourier Expansions
7.10 Asymptotic Series
8. Differential Equations
8.1 First-Order Differential Equations.
8.2 AC Circuits
8.3 Second-Order Differential Equations
8.4 Some Examples from Physics
8.5 Boundary Conditions.
8.6 Series Solutions
8.7 Bessel Functions.
8.8 Second Solution
9. Matrix Algebra
9.1 Matrix Multiplication
9.2 Further Properties of Matrices
9.3 Determinants.9.4 Matrix Inverse
9.5 Wronskian Determinant
9.6 Special Matrices.
9.7 Similarity Transformations.
9.8 Eigenvalue Problems.
9.9 Group Theory9.10 Minkowski Spacetime
10. Multivariable Calculus
10.1 Partial Derivatives
10.2 Multiple Integration
10.3 Polar Coordinates.
10.4 Cylindrical Coordinates
10.5 Spherical Polar Coordinates
10.6 Differential Expressions.
10.7 Line Integrals
10.8 Green’s Theorem
11. Vector Analysis
11.1 Scalars and Vectors
11.2 Scalar or Dot Product
11.3 Vector or Cross Product.
11.4 Triple Products of Vectors.
11.5 Vector Velocity and Acceleration.
11.6 Circular Motion
11.7 Angular Momentum
11.8 Gradient of a Scalar Field
11.9 Divergence of a Vector Field.
11.10 Curl of a Vector Field.
11.11 Maxwell’s Equations.
11.12 Covariant Electrodynamics.
11.13 Curvilinear Coordinates.
11.14 Vector Identities
12. Partial Differential Equations and Special Functions
12.1 Partial Differential Equations.
12.2 Separation of Variables
12.3 Special Functions.
12.4 Leibniz’s Formula
12.5 Vibration of a Circular Membrane.
12.6 Bessel Functions.
12.7 Laplace’s Equation in Spherical Coordinates.
12.8 Legendre Polynomials.
12.9 Spherical Harmonics.
12.10 Spherical Bessel Functions.
12.11 Hermite Polynomials.
12.12 Laguerre Polynomials
13. Complex Variables
13.1 Analytic Functions.
13.2 Derivative of an Analytic Function.
13.3 Contour Integrals
13.4 Cauchy’s Theorem
13.5 Cauchy’s Integral Formula.
13.6 Taylor Series
13.7 Laurent Expansions
13.8 Calculus of Residues
13.9 Multivalued Functions
13.10 Integral Representations for Special Functions
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