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integral – Kepler, Johannes

integral   An antiderivative of a function. That is, if f(x) is a real-valued function, an antiderivative F(x) of f(x) has the property that the derivative of F with respect to x is f.
The definite integral (called the Riemann integral) of a real-valued function f(x) from x = a to x = b is the limit of the Riemann sum:

assuming this limit exists, where ci is in the i th subinterval of the partition of (a, b), and where a is the lower limit, and b is the upper limit, of the integral.

Cf. fundamental theorem of calculus, Lebesgue integral.

integral test   A test for the convergence of a series. See the related article for a complete description.

Related article: Series

integration   Obtaining an integral of a function.

integration formulas   See the article for a complete list of common integration formulas.
Cf. integral.

interior   The interior of a subset E of a topological space is the largest open set contained in E. It may also be expressed as the intersection of E with the closure of its complement. If E is open, then it is equal to its interior. If E contains no open sets we say that it has empty interior, and this equivalent to saying that E is nowhere dense.

intersection   The intersection of two sets A and B is the set of those elements common to both A and B, and is denoted by

Thus, an element x belongs to the intersection only if it is an element of A and also an element of B. If the intersection is taken over a family of sets {Ai}i = 1, 2, ... n, then it is the set of those elements that are in every set in the family, denoted

Sometimes one speaks of the interesection over a (single) set A, and this indicates the set of elements that are in every element of A:

When considering an algebra of sets, the intersection of two or more sets is sometimes called the meet of the sets.
Cf. union.

interval   The set of all real numbers lying between two given real numbers a and b. If both a and b are included in the set it is called a closed interval. If neither a nor b is included it is called an open interval. If only one of a or b is included the interval is called half-open (equivalently, half-closed). Intervals may be denoted using either interval notation or set notation, as follows:

Although intervals are most commonly intervals on the real line, the definition carries over without modification to any totally ordered set.

interval graph   An intersection graph of a finite family of intervals on the real line.

inverse function   If f is an injective (i.e., one-to-one) function, then f is said to be invertible, and its inverse is the function g satisfying g(f(x)) = f(g(x)) = x for every x in the domain of f.

inverse image   Given a function f and a subset Y of the range of f, the inverse image (under f) of Y is the set of all x such that f(x) is in Y.

inverse statement   Given a conditional statement, i.e., a statement of the form “if A then B,” or “A implies B,” its inverse is “not A implies not B.” A conditional neither implies nor is implied by its inverse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other.

irrational number   A real number which is not a rational number, i.e., which cannot be expressed as a ratio of integers. Examples: p (the ratio of the circumference of a circle to its diameter) and the square root of 2 (ratio of the length of the diagonal of a square to the length of one side). The irrational numbers are uncountably infinite.

Related article: Irrationality of the Square Root of 2
Related MiniText: Number -- What Is How Many?

irreflexive   See reflexive relation.

isomorphism   A morphism that is both injective and surjective, that is, both “one-to-one” and “onto.” An isomorphism from a structure to itself is called an automorphism.

isotone function   A function that is order preserving and increasing.

join   A binary operation whose value on two elements a and b of a lattice is the least upper bound of a and b, denoted a b.
Cf. meet.

join irreducible   An element a of a lattice is called join irreducible if whenever a = b c then a = b or a = c. More generally, a is called strictly join irreducible if whenever a is the join of a subset X of the lattice, then a is an element of X.
Cf. join prime, meet irreducible, meet prime.

join-morphism   A function f on a lattice L is called a join-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a join-morphism, then f is called a join-isomorphism. A join-isomorphism from a lattice to itself is called a join-automorphism.
Cf. join, meet-morphism.

join prime   An element a of a lattice is called join prime if whenever a b c then a b or a c. Also, a is called strictly join prime if

Cf. join irreducible, meet prime, meet irreducible.

Jordan Decomposition Theorem   If m is a signed measure, there exist unique positive measures m+ and m such that m = m+m, with m+ and m mutually singular. This is called the Jordan decomposition of m, and the measures m+ and m are called the positive and negative variations of m. The total variation of m is defined to be the sum of the positive and negative variations of m.

 Johannes Kepler
Kepler, Johannes
(born 1571)   German astronomer and mathematician who discovered that planetary motion is elliptical. Early in his life, Kepler set about to prove that the universe obeyed Platonistic mathematical relationships; that for instance the planetary orbits were circular and at distances from the sun proportional to the Platonic solids. However, when his friend the astronomer Tycho Brahe died he bequeathed to Kepler his immense collection of astronomical observations. After decades of studying these observations, Kepler realized that his earlier surmises about planetary motion were naive, and he formulated his three laws of planetary motion, now known as Kepler's Laws. He did not have a unifying theory for these laws, however; this had to wait another generation, until Isaac Newton formulated his laws of gravity and motion.

Related article: Kepler's Laws

integral – Kepler, Johannes

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