Mathematics-I

Created by Pavl_G

Outline:

This repository houses re-usable equations and formulas for the following:

• Calculus

  • Functions.
  • Transcendental functions.
  • Differential calculus.
  • The Chain Rule.
  • Fundamental theorem of calculus.
  • Integration by parts (anti-derivation of the chain rule).
  • First-order differential equations.
  • Second-order differential equations.
  • Vectors.
  • Basic operations on vectors.
  • Dot product and projection vectors.
  • Cross product.
  • Triple product.
  • Polar coordinates.
  • Polar-to-rectangular coordinates.
  • Vectorial analysis using the fundamental theorem.
  • Appendices: Proofs
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  • Appendix-A: Derivation of the first-order differential equations for the transcendental functions.
  • Appendix-B: Derivation of the trigonometric identities.
  • Appendix-C: Derivation of the fundamental theorem part-01 & part-02.
  • Appendix-D: Derivation of the vector normalization formula.
  • Appendix-E: Derivation of the dot product and the projection vectors.
  • Appendix-F: Derivation of the cross product.
  • Appendix-G: Derivation of the triple product.
  • Appendix-H: Derivation of the vectorial polar coordinates from the rectangular coordinates.
  • Appendix-I: Mathematical analysis models (univariable & multivariable), and continuum to Mathematics-II.
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• Discrete Mathematics

  • Sets & Relations.
  • Types of relations: Paths. Operations. Undetermined relations.
  • Finite-State-Automaton.
  • Mapping FSA using Sets and Relations.
  • Manipulating relations in computerized data structures.
  • Mapping FSA using vectors and matrices data structures.
  • Algorithms on FSA.
  • Asymptotic analysis on FSA.
  • The fundamental theorem of calculus in FSA-based algorithms.
  • Appendices
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  • Appendix-A: Symbolic Designation of mathematical analysis
  • Appendix-B: Algorithm Analysis
  • Appendix-C: Analysis by example
  • Appendix-D: FSA Analysis.
  • Appendix-F: Time-warp states and FSA.
  • Appendix-E: Implementation using Automata4j.
  • =============================

• Linear algebra

  • Elimination methods.
  • Gaussian’s Elimination.
  • Vectorial matrices.
  • Linear dependance and independence.
  • Linear Interpolation.
  • Linear Extrapolation.
  • Eigenvalues.
  • Appendices
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  • WIP
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