Mathematics of theoretical physics

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Mathematics is a very important tool in physics. In many ways mathematics could be considered the language through which physical theories and relations are expressed. Now different branches of mathematics happen to be relevant to different areas of physics but many mathematical tools can be applied to a wide class of problems.

Basic Mathematics[edit]

These areas are critical to understand before moving on to more advanced mathematics or before attempting to learn physics at anything above a conceptual level. This would be something close to the mathematical requirements for high school level physics courses.

  • Algebra
  • Trigonometry
  • Geometry
  • Single Variable Calculus

Multi-variable Calculus[edit]

Main Article: Multivariable Calculus
Multi-variable calculus is concerned with functions of more than a single variable, and extends itself to vector analysis. It is essential in the understanding of continuous theories and forms a basis of a great many topics in physics, most notably electro-magnetic theory and Maxwell's equations.

Differential Equations[edit]

Differential Equations relate derivatives of functions, and are used to model systems and problems in almost all continuous theories. Again it is an essential topic for a anyone interested in pursuit of a quantitative understanding of theoretical physics. Applications of this topic can be found in almost any topic in physics. A few examples include:

  • Thermodynamics - Heat Equation
  • Quantum Mechanics - Wave Equation (Schrodinger Equation)
  • Lagrangian Mechanics - Equations of Motion
    • Action and Symmetry, Conservation Laws - Noether's Current

Linear Algebra[edit]

Group Theory[edit]

Differential Geometry[edit]

Miscellaneous Mathematical Tools[edit]

  • Legendre Transformation: Very useful in thermodynamics.