Vector and Other Analyses
What are some other types of analysis?
There are numerous other analyses in mathematics and the sciences besides vector analysis. And each one has applications that sometimes intermingle but most often has specific uses in certain areas—from computer programming to electronics.
At one time, the study of tensors was known as the absolute differential calculus, but today it is simply called tensor analysis. Tensors were originally invented as the extensions of vectors. Tensor analysis is concerned with relations or laws that remain valid regardless of the coordinate system used to specify the quantities.
Complex analysis (or complex variable analysis) is the study of complex numbers and their derivatives, mathematical manipulations, and other properties. It is mostly used to find the solution to holomorphic functions, or those that are found in a complex plane, use complex values, and are differentiable as complex functions. Complex variables deal with the calculus of functions of a complex variable, incorporating differential equations and complex numbers; for example, one such variable of the form z = x + iy, in which x and y are real and the imaginary number i = -1. Complex-variable techniques have a great many uses in applied areas, such as electromagnetics.
Functional analysis is concerned with infinite-dimensional vector spaces and the mapping between them. It is also considered the study of spaces of functions.
There is also differential geometry, truly a branch of geometry, which includes the concepts of the calculus as applied to curves, surfaces, and other geometry entities. Originally, it included the use of coordinate geometry; more recently, it has been applied to other areas of geometry, such as projective geometry. In particular, differential geometry uses techniques of differential calculus to determine the geometric properties of manifolds (a topological space that resembles Euclidean space, but is not).
Another type of analysis is nonstandard analysis, developed in the 1960s, in which hyperreal numbers are used to define the existence of “genuine infinitesimals.” (These are numbers less than one half, one third, one fourth, and so on, but greater than 0.) It is used in several fields, including probability theory and mathematical physics.