glossary page 89
a^3: * see Endnote 83
dimension of a mathematical space (or object) is defined as the minimum number of coordinates to specify any point within it. A line has a dimension of 1 because 1 coordinate is needed to specify a point on it (the point at 5 on a number line). A surface such as a plane or sphere has a dimension of 2 as 2 coordinates are needed to specify a point on it (latitude & longitude are required to locate a point on the surface of a sphere). The inside of a cube is 3 dimensional as 3 coordinates are needed to locate a point within these spaces.
a^n: * see Endnote 84
as an abstract dimension (analytic but not visual); dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics & sciences, as parameter spaces or configuration spaces (eg Lagrangian or Hamiltonian mechanics); these are abstract spaces, independent of the physical space we live in.
hypercomplex:
as an element of a finite-dimensional algebra over the real numbers that is unital and distributive (but not necessarily associative);elements are generated with real number coefficients
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quaternions of the calculus of vectors:
a number system that extends the complex numbers, first described by WR Hamilton in 1843, applied to mechanics in 3D space; a feature of quaternions is that multiplication of two quaternions is noncommutative; Hamilton defined a quaternion as the quotient of two directed lines in a 3D space or equivalently as the quotient of 2 vectors.
any finite number raised to the power of infinity (means that we are multiplying that finite number infinite times) is equal to Infinity .
2^ ∞ = 2 x 2 x2 ...........= ∞ If that finite number is 1, then we can say that it is indeterminate form. In calculus (and other branches of mathematical analysis), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is an indeterminate form.
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subtilizing:
to elevate in character; sublimate; to make (the mind, senses, etc.) keen or discerning; sharpen; to introduce subtleties into or argue subtly about; to make thin, rare, or more fluid or volatile; refine.
groups: * see Endnote 85
an algebraic structure consisting of a set of elements plus an operation which when applied to any two elements forms a third element; which satisfies 4 group axioms: closure, associativity, identity & invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation.
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sets:
(aka aggregates) a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored. Members of a set are often referred to as elements and the notation a in A is used to denote that a is an element of a set A. The study of sets and their properties is the object of set theory. It is a collection of distinct objects, each object considered as an object in its own right (e.g. numbers 2, 4, & 6 are distinct objects considered separately, considered collectively they form a single set of size three, written {2,4,6})
homogenous:
relating to a function of several variables that becomes multiplied by some power of a constant when each variable is multiplied by that constant: x^ 2y^ 3 -a homogeneous expression of degree 5.
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mathematical images: * see Endnote 86
the subset of a function's codomain which is the output of the function from a subset of its domain.
Evaluating a function at each element of a subset X of the domain, produces a set called the image of X under or through the function.
totality:
the whole amount, the state of being total
differential equations: * see Endnote 87
relates some function with its derivatives; the functions usually represent physical quantities, the derivatives represent their rates of change, & the equation defines a relationship between the two.
