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 $\otimes$, $\odot$
- Notation of expectation values in physics $\braket{x}$
- 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 $f = f(x,y,z)$ and vector functions $\mathbf{f}(x,y,z) = \left( f_{1}, f_{2}, f_{3} \right)$. However, in physics, the term vector is often used simply instead of vector function.
Common notations for scalar functions in physics include $T, V, U, \phi, \psi$, etc. Scalar functions are also called scalar fields.
$$T = T(x,y,z)$$
A function is represented mathematically as $T(x,y,z)=2xy+z^{2}$, with temperature being a concrete example. Given any coordinate $(x,y,z)$ 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 $\mathbf{A}, \mathbf{B}, \mathbf{v}$, etc. Vector functions are also called vector fields.
$$ \mathbf{A} = \mathbf{A}(x,y,z) = (A_{x}, A_{y}, A_{z}) = A_{x}\hat{\mathbf{x}} + A_{y}\hat{\mathbf{y}} + A_{z}\hat{\mathbf{z}} $$
A function is represented mathematically as $\mathbf{A}(x,y,z) = \left( xy, 2y^{2}, 3xyz \right) = xy\hat{\mathbf{x}} + 2y^{2}\hat{\mathbf{y}} + 3xyz\hat{\mathbf{z}}$, with velocity being a concrete example. For an object moving in three-dimensional space at a coordinate $(x,y,z)$, 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 $t$. In this case, it is expressed as a univariate vector function form. If $\mathbf{x}(t) = (x_{1}(t), x_{2}(t), x_{3}(t) )$, then,
$$ \mathbf{A} (t) = \mathbf{A}(\mathbf{x}(t)) = \mathbf{A}(x_{1}(t), x_{2}(t), x_{3}(t) ) = \left( A_{1}(t), A_{2}(t), A_{3}(t) \right) $$
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 $\mathbf{R} = \mathbf{r} - \mathbf{r}^{\prime}$
- 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
- Notation of Expectation Values in Physics
- 방향각과 방향코사인
- 두 벡터의 내적과 사잇각의 관계
- 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