culus, 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 com-
plex plane, use complex values, and are differentiable as complex functions. Complex
variables deal with the calculus of functions of a complex variable, incorporating differ-
ential equations and complex numbers; for example, one such variable of the form z
xiy,in which xand yare real and the imaginary number i1. Complex-variable
techniques have a great many uses in applied areas, such as electromagnetics.
Functional analysisis concerned with infinite-dimensional vector spaces and the
mapping between them. It is also considered the study of spaces of functions. Finally,
differential geometryis really a branch of geometry, but it includes the concepts of
the calculus as applied to curves, surfaces, and other geometric 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 man-
ifolds (a topological space that resembles Euclidean space, but is not).
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MATHEMATICAL ANALYSIS
What is a linear combination?
A
linear combination is a combination or the sum of two or more entities with
each multiplied by some number (with not all the numbers being zero). Lin-
ear combinations of vectors, equations, and functions are common. For exam-
ple, if xand yare vectors and aand bare numbers, then axbyis a linear com-
bination. (For more information about equations and functions, see “Math
Basics” and “Algebra.”)