Mathematical Physics
Gamma function, Bessel function, Legendre polynomials and other special functions can be found in the ‘Functions’ category.
Basic
- What is a sufficiently small angle?
- General definition of potential, potential energy
- What is a differential operator?
- Notation for vectors entering/leaving a plane ,
- Notation of expectation values in physics
- What is flux in physics?
Coordinate Systems
- Coordinates and Coordinate Systems
- Number Line
- Coordinate Plane
- Coordinate Space, Cartesian Coordinate System
- Polar Coordinate System
- Cylindrical Coordinate System
- Spherical Coordinate System
Vector Analysis
While slightly less rigorous mathematically, this section deals with vector analysis at a level intended for physics and engineering majors, limited to three dimensions. General multivariable analysis and vector analysis are covered in the multivariable vector analysis category.
Vector analysis discusses scalar functions and vector functions . However, in physics, the term vector is often used simply instead of vector function.
Common notations for scalar functions in physics include , etc. Scalar functions are also called scalar fields.
A function is represented mathematically as , with temperature being a concrete example. Given any coordinate in three-dimensional space, the temperature at that location is a scalar value, so temperature is represented by a scalar function.
Common notations for vector functions include , etc. Vector functions are also called vector fields.
A function is represented mathematically as , with velocity being a concrete example. For an object moving in three-dimensional space at a coordinate , its velocity at that location is a three-dimensional vector, so velocity is represented by a vector function.
However, in physics, the variables of a vector function often depend on time . In this case, it is expressed as a univariate vector function form. If , then,
Vector Algebra
- What is the dot product in 3-dimensional space
- What is the cross product in 3-dimensional space
- Vector triple product, BAC-CAB formula
- Scalar triple product
- What is a pseudovector?
Vector Differentiation
- Differentiation of vectors, dot product, and cross product in Cartesian coordinates
- Derivatives of 3-dimensional scalar/vector functions
- Separation vector
- What is the del operator in physics
- Differentiating the Heaviside step function yields the Dirac delta function
Vector Integration
- Fundamental theorem of gradients
- Gauss’s theorem, Divergence theorem
- Stokes’ theorem
- Partial integration of expressions involving the del operator
- Various formulas for vector integration involving the del operator
- Pappus-Guldinus theorem
- Definition and properties of vector area
Tensor Analysis
Tensor
Curvilinear Coordinates
- Curvilinear coordinate systems in 3-dimensional space
- Scale factors
- Coordinate transformation and Jacobian
- Del operator in curvilinear coordinates
- Formulas for gradient, divergence, curl, and Laplacian
Differential Equations
- Basics of differential equations
- Spherical harmonics: General solution for the polar and azimuthal angles in the Laplace equation in spherical coordinates
- Solution of Laplace’s equation independent of azimuthal angle in spherical coordinates using the method of separation of variables
- Solution of Laplace’s equation independent of the z-axis in cylindrical coordinates using the method of separation of variables
All posts
- Vector Triple Product, BAC-CAB Rule
- Levi-Civita Symbol
- Kronecker Delta
- Einstein Notation
- Product of Two Levi-Civita Symbols
- Gradient of the Magnitude of Separation Vectors
- Separation Vector
- Scalar Triple Product
- Unit Vectors of the Spherical Coordinate System Expressed in Terms of Unit Vectors of the Cartesian Coordinate System
- Euclidean Space
- The Curl of a Gradient is Always Zero
- Curl of the Curl of Vector Functions
- The Divergence of Curl is Always Zero
- Cross Product in Three-Dimensional Euclidean Space
- Conversion of Cartesian Coordinate System Unit Vectors to Spherical Coordinate System Unit Vectors
- Gradient, Divergence, Curl, and Laplacian in Curvilinear Coordinates
- Differentiation of Vectors, Dot Product, and Cross Product in Cartesian Coordinates
- Gauss's Theorem, Divergence Theorem
- What is a Pseudovector?
- The Fundamental Theorem of Slopes
- Divergence of a Separation Vector
- Solution of the Laplace Equation Independent of Azimuthal Angle in Spherical Coordinates using the Method of Separation of Variables
- Solving the Laplace Equation Independent of the Jet Axis in Cylindrical Coordinates Using the Method of Separation of Variables
- Equations Involving the Laplacian Operator, Second Order Partial Derivatives
- Proof that Differentiating the Heaviside Step Function Yields the Dirac Delta Function
- Rotation of Separation Vector
- Stokes' Theorem
- Partial Integration of Expressions Containing the Del Operator
- In Physics, What is a Tensor
- Derivation of Bessel's Equation
- Physics Appendix
- For an Angle Small Enough
- Differential Equations for Physics: Solutions to Commonly Encountered Differential Equations
- Physics에서의 Del 연산자
- Spherical Harmonics: General Solutions for the Polar and Azimuthal Angles in the Spherical Coordinate Laplace's Equation
- Normalization of Spherical Harmonic Functions
- What is a Differential Operator in Physics?
- Operator Solution of the Differential Equation Satisfied by Hermite Functions
- The Magnitude of the Cross Product of Two Vectors is Equal to the Area of the Parallelogram They Form
- Equation of an Ellipse with the Focus at the Origin in Polar Coordinates
- Elliptic Integral of the Second Kind
- Curl of Vector Functions in 3D Cartesian Coordinates
- Differential Volume in Cylindrical Coordinates
- Infinitesimal Area in Polar Coordinates, Infinitesimal Volume in Cylindrical Coordinates
- Gradient of Scalar Function in Cartesian Coordinate System
- Divergence of Vector Function in Cartesian Cooridenates System
- Reasons Not to Use r, Theta as Variables in Cylindrical Coordinates
- Total Differentiation, Exact Differentiation
- Curvilinear Coordinates in Three-Dimensional Space
- Scaling Factors of Curvilinear Coordinates
- Curved Coordinate Systems: Coordinate Transformations and Jacobians
- Gradient of a Scalar Function in Curvilinear Coordinates
- Divergence of Vector Functions in Curvilinear Coordinates
- Laplacian of a Scalar Function in the Three-Dimensional Cartesian Coordinate System
- Laplacian of a Scalar Function in Curvilinear Coordinates
- Product Rule Involving the Del Operator
- Various Formulas of Vector Integration Involving the Del Operator
- Definition and Properties of Vector Areas
- Derivatives of 3D Scalar/Vector Functions
- Potential, A General Definition of Potential Energy
- Lagrange Multiplier Method
- Coordinate Systems and Coordinates in Physics
- Physics에서 Coordinate Transformation
- What is the Dot Product in Three-Dimensional Space?
- Curl of a Vector Function in Curvilinear Coordinates
- Notation for Vectors Entering and Exiting the Plane
- Definition of a Line
- Definition of Coordinate Plane
- Polar Coordinate System
- Coordinate Space, Cartesian Coordinate System
- Sperical Coordinate System
- What is Flux in Physics?
- Gradient, Divergence and Curl in Curvilinear Coordinates
- Curvilinear Coordinate System and the Del Operator
- Cylindrical Coordinates에서의 Del 연산자
- Cylindrical Coordinate System
- Notation of Expectation Values in Physics
- Azimuth and Direction Cosines
- The Relationship between the Dot Product of Two Vectors and the Angle Between Them