logo

Mathematical Physics

Gamma function, Bessel function, Legendre polynomials and other special functions can be found in the ‘Functions’ category.

Basic

Coordinate Systems

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

Vector Differentiation

Vector Integration

Tensor Analysis

Tensor

Curvilinear Coordinates

Differential Equations


All posts