Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Mathematical Physics


Mathematical physics is an interdisciplinary field that combines mathematics with physics. It involves the application of mathematical methods by which one can discover, explain, and often solve problems that arise from the physical world. For graduate students studying mathematics, it offers a unique perspective on how abstract mathematical theories can be integrated into the more tangible field of physics.

The study of mathematical physics bridges the gap between theoretical physics and mathematical rigor. This discipline can uncover the mathematical structures that underlie physical theories. One way to think of it is as a form of mathematical modeling, where concepts such as differential equations, linear algebra, and tomography play important roles. It is about using mathematics to express the operation of the universe.

Historical context

Mathematics and physics have been intertwined for centuries. From Euclid's early works on geometry to Newton's Principia Mathematica, mathematics has served as a language for expressing physical phenomena. The laws governing motion, gravity, and thermodynamics all derive from mathematical models. As science progressed, the mathematical complexity required to accurately describe the universe also increased.

Baseline concepts

Calculus in physics

Calculus is at the heart of physics because it enables scientists to describe change. Whether it is the trajectory of a missile or the growth rate of a bacterial colony, calculus provides the tools to model these processes.

Example: The basic formula for velocity in calculus terms is:

v(t) = frac{d}{dt} s(t)

where v(t) is velocity, s(t) is a function of position relative to time t.

Differential equations

Differential equations are perhaps the most important part of mathematical physics. These equations involve functions and their derivatives, which represent physical quantities and their changes over time.

Example: Newton's second law:

F = m cdot a

can be expressed using a differential equation. If you express the acceleration a as the second derivative of the position function s(t), you get:

F = m cdot frac{d^2}{dt^2} s(t)

Mathematical methods

Linear algebra

Linear algebra is another fundamental mathematical tool used in physics. It involves the study of vectors, vector spaces, and linear transformations, which are important in the explanation of quantum mechanics, electromagnetism, and general relativity.

Visual example: Subjecting points on a plane represented by vectors to transformations such as rotations can be described using matrix multiplication.

Partial differential equations

While ordinary differential equations deal with functions of a single variable, partial differential equations (PDEs) involve functions of several variables and their partial derivatives. These equations are fundamental in describing phenomena such as heat conduction, wave propagation, and quantum mechanics.

Example: The heat equation in one dimension is a PDE:

frac{partial u}{partial t} = alpha frac{partial^2 u}{partial x^2}

where u is the temperature distribution function, t is time, x is the spatial coordinate, and alpha is the thermal diffusion constant.

Theory and applications

Quantum mechanics

Quantum mechanics has fundamentally changed our understanding of the universe at the microscopic scale. The main aspects of quantum mechanics are described mathematically using linear algebra and special functions.

Example: The basis of quantum mechanics, the Schrödinger equation, uses partial differential equations to describe the wave function of particles:

ihbar frac{partial}{partial t} Psi = hat{H} Psi

Einstein's general theory of relativity

Einstein's theory, which describes the force of gravity, uses advanced calculus and algebra. It argues that massive objects create distortions in spacetime, which are felt as gravity.

Example: The general relativity field equations are non-linear PDEs:

R_{munu} - frac{1}{2}g_{munu}R + g_{munu}Lambda = frac{8pi G}{c^4}T_{munu}

Notable techniques

Fourier analysis

Fourier analysis is used to decompose functions into sinusoidal components. It is particularly useful in signal processing and solving PDEs.

Example: Breaking down a complex sound wave into simpler sine waves makes it easier to analyze.

Complex analysis

Complex analysis studies functions that operate on complex numbers, which appear in various aspects of mathematical physics. For example, complex analysis is used in the evaluation of certain integrals.

Example: Complex numbers can simplify many mathematical expressions, such as in electromagnetism.

z = x + yi

Conclusion

Mathematical physics is a vast field that connects abstract mathematical theories to concrete physical phenomena. By understanding the mathematical tools and principles that govern physics, students not only learn to solve fascinating problems but also gain deeper insights about the universe. This field is an exciting journey where the abstract beauty of mathematics meets the complex workings of the physical world.


Undergraduate → 12


U
username
0%
completed in Undergraduate

← Prev (11.4)
Projective Geometry
Next (12.1) →
Fourier Series

Comments 



Mathematical Physics

https://www.buddymath.com/ug/12?bmal=en