How to become a
GOOD Theoretical Physicist

Primary Mathematics

Now, first things first. Are you comfortable with numbers, adding, subtracting, square roots, etc.?

• Natural numbers: 1, 2, 3, …
• Integers: …, -3, -2, -1, 0, 1, 2, …
• Rational numbers (fractions): $\dfrac{1}{2}$, $\dfrac{1}{4}$, $\dfrac{3}{4}$, $\dfrac{23791}{773}$, $\dots$
• Real numbers: Sqrt(2) = 1.4142135… , π = 3.14159265… , e = 2.7182818…, …
• Complex numbers: 2+3i, e^ia= cos(a) + i sin( a), … they are very important!
• Set theory: open sets, compact spaces. Topology.You may be surprised to learn that they do play a role indeed in physics!
• Algebraic equations. Approximation techniques. Series expansions: the Taylor series.
• Solving equations with complex numbers. Trigonometry: sin(2x)=2sin x cos x, etc.
• Infinitesimals. Differentiation. Differentiate basic functions (sin, cos, exp).
• Integration. Integrate basic functions, when possible. Differential equations. Linear equations.
• The Fourier transformation. The use of complex numbers. Convergence of series.
• The complex plane. Cauchy theorems and contour integration (now this is fun).
• The Gamma function (enjoy studying its properties).
• Gaussian integrals. Probability theory.
• Partial differential equations. Dirichlet and Neumann boundary conditions.

This is for starters. Some of these topics actually come as entire lecture courses. Much of those are essential ingredients of theories in Physics. You don’t have to finish it all before beginning with what follows next, but remember to return to those subjects skipped during the first round.

• Beginning Algebra
• Intermediate Algebra
• Dave E. Joyce's Trigonometry course
• Prof. James Binney's course on Complex Numbers (*.pdf)
• (Nearly) Complete overview of Primary Mathematics (K. Kubota, Kentucky)
• Chris Pope's lecture notes: Methods 1  Methods 2
• The complex plane, Cauchy theorems and contour integration.