Calculate derivatives, integrals and Taylor series. Use Euler's formula. Understand the properties of matrices. Compute determinants and matrix diagonalization. Find general solutions of linear ordinary differential equations. Understand how to solve simultaneous ordinary differential equations. Determine equations of motion for coupled oscillations and obtain solutions. Compute partial derivatives. Compute line integrals and surface integrals. Prove Green's theorem. Compute gradients, divergences, and rotations. Use the ε tensor to prove formulas for vector analysis. Compute line integrals, surface integrals, and volume integrals. Use Gauss's divergence theorem and Stokes' theorem. Compute Cauchy-Riemann equation. Understand Cauchy's integral theorem. Compute residue integrals. Find the solutions of coupled oscillations under periodic boundary conditions. Compute Fourier series. Compute Fourier transform. Understand the properties of delta functions. Compute delta functions using Fourier transform. Find the general solutions of one dimensional wave equation. Find the solutions of Poisson's equation using Green's function. Compute solutions of Feynman kernel for simple examples. Find the Green's function of three dimensional wave equation. Understand variational methods. Solve the simple problems using variational methods. Understand the properties of group theory. Construct the representation of su(2). Compute spherical harmonic function.
Outline:
The most important thing in physics is the physical concepts, and mathematics is merely a tool. However, if one does not understand how to use the mathematical tools, it is impossible to grasp the physical concepts. In this lecture, the goal is to learn mathematics, which is indispensable for physics, by focusing on concrete examples. Some students may find it difficult to proceed to the next step because of difficult mathematical terms and rigorous proofs when they reach for specialized mathematical books. In this lecture, we will not go into rigorous proofs. The exercises will be based on past university entrance examinations.
Style:
Students are expected to speak up actively in the lectures and ask questions that they do not understand or have questions about. Also, actively discuss with your friends and seniors. The cycle of preparation→lecture→review is important so that students can quantitatively understand their level of understanding. The textbook is very carefully written, so students should read the textbook by themselves and try to fill in the gaps between the lines. The students should study various reference books on how to solve the exercises and learn how to solve the problems in a way that they can understand.
【Lecture:60 hours】
Notice:
This lecture is intended for students who wish to enter university or advanced courses. Other students who are interested in mathematics and physics, and who can follow the difficult contents with patience, are also eligible for this course. Students are required to review the mathematics and physics they have learned in the past.
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Theme |
Goals |
| 1st Semester |
| 1st Quarter |
| 1st |
Derivatives, integrals and complex numbers |
Calculate derivatives, integrals and Taylor series. Use Euler's formula.
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| 2nd |
Matrices, determinants and matrix diagonalization |
Understand the properties of matrices. Compute determinants and matrix diagonalization.
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| 3rd |
Linear ordinary differential equation |
Find general solutions of linear ordinary differential equations.
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| 4th |
Simultaneous ordinary differential equations |
Understand how to solve simultaneous ordinary differential equations.
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| 5th |
Coupled oscillators |
Determine equations of motion for coupled oscillations and obtain solutions.
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| 6th |
Two-variable functions and partial differentiation |
Compute partial derivatives.
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| 7th |
Integration of 2D vector analysis |
Compute line integrals and surface integrals.
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| 8th |
1st semester midterm examination |
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| 2nd Quarter |
| 9th |
Green's theorem |
Prove Green's theorem.
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| 10th |
gradient, divergence and rotation |
Compute gradients, divergences, and rotations. Use the ε tensor to prove formulas for vector analysis.
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| 11th |
Integration of 3D vector analysis |
Compute line integrals, surface integrals, and volume integrals.
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| 12th |
Gauss's divergence theorem and Stokes's theorem |
Use Gauss's divergence theorem and Stokes' theorem.
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| 13th |
Holomorphic function |
Compute Cauchy-Riemann equation.
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| 14th |
Integrals of holomorphic functions |
Understand Cauchy's integral theorem.
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| 15th |
Isolated singularity and residue integral |
Compute residue integrals.
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| 16th |
1st semester final examination |
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| 2nd Semester |
| 3rd Quarter |
| 1st |
Discrete Fourier transform |
Find the solutions of coupled oscillations under periodic boundary conditions.
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| 2nd |
Periodic function and Fourier series |
Compute Fourier series.
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| 3rd |
Fourier transform |
Compute Fourier transform.
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| 4th |
Delta function (1) |
Understand the properties of delta functions.
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| 5th |
Delta function (2) |
Compute delta functions using Fourier transform.
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| 6th |
One dimensional wave equation |
Find the general solutions of one dimensional wave equation.
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| 7th |
Laplace's equation and Poisson equation |
Find the solutions of Poisson's equation using Green's function.
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| 8th |
2nd semester midterm examination |
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| 4th Quarter |
| 9th |
Schrodinger equation |
Compute solutions of Feynman kernel for simple examples.
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| 10th |
Three dimensional wave equation and Green's function |
Find the Green's function of three dimensional wave equation.
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| 11th |
Introduction of variational methods |
Understand variational methods.
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| 12th |
Applications of variational methods |
Solve the simple problems using variational methods.
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| 13th |
Symmetry and physics |
Understand the properties of group theory.
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| 14th |
Representation of su(2) |
Construct the representation of su(2).
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| 15th |
Spherical harmonic function |
Compute spherical harmonic function.
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| 16th |
2nd semester final examination |
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