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?

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
  • The magnitude of the cross product of two vectors equals the area of the parallelogram they form
• 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 $\mathbf{R} = \mathbf{r} - \mathbf{r}^{\prime}$
  • Gradient of magnitude $\nabla (\mathbf{R}^n)=n\mathbf{R}^{n-1}\hat{\mathbf{R}}$
  • Divergence $\nabla \cdot \left( \dfrac{1}{\mathbf{R}^{2}} \hat{\mathbf{R}} \right) =\ 4\pi \delta^3(\mathbf{R})$
  • Curl $\nabla \times \dfrac{\hat{\mathbf{R}}}{\mathbf{R}^2} = \mathbf{0}$
• What is the del operator in physics
  • Gradient of a scalar function
  • Divergence of a vector function
  • Curl of a vector function
  • Laplacian of a scalar function
  • Multiplication rules involving the del operator
  • Second-order derivatives involving the del operator twice
    • The curl of a gradient is always $\mathbf{0}$
    • The divergence of a curl is always 0
    • The curl of the curl of a vector function
• 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

Coordinate Systems

• Coordinates and Coordinate Systems
• Number Line
• Coordinate Plane
• Coordinate Space, Cartesian Coordinate System
• Polar Coordinate System
  • Equation of an Ellipse with a Focus at the Origin in Polar Coordinates
  • Area Element in Polar Coordinates, Volume Element in Cylindrical Coordinates
• Cylindrical Coordinate System
  • Why We Shouldn’t Use $r$, $\theta$ as Variables in Cylindrical Coordinates
• Spherical Coordinate System
  • Unit Vectors in Spherical Coordinates Expressed with Cartesian Unit Vectors
  • Expressing Cartesian Unit Vectors in Terms of Spherical Coordinate System Unit Vectors
  • Volume Element in Spherical Coordinates

Tensor Analysis

Tensor

• What is a tensor in physics
• Einstein notation
• Kronecker delta
• Levi-Civita symbol
  • Product of two Levi-Civita symbols

Curvilinear Coordinates

• Curvilinear coordinate systems in 3-dimensional space
• Scale factors
• Coordinate transformation and Jacobian
• Del operator in curvilinear coordinates
  • Del operator in cylindrical coordinates
  • Del operator in spherical coordinates
• Formulas for gradient, divergence, curl, and Laplacian
  • Gradient of a scalar function
  • Divergence of a vector function
  • Curl of a vector function
  • Laplacian of a scalar function

Differential Equations

• Basics of differential equations
• Spherical harmonics: General solution for the polar and azimuthal angles in the Laplace equation in spherical coordinates
  • Normalization of spherical harmonics
• 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