Concepts about Calculus
Swiss family that produced many mathematicians and scientists in the 17th, 18th, and 19th centuries, in particular the brothers Jakob (1654-1705) and Johann (1667-1748).
Jakob and Johann were pioneers of Leibniz’s calculus. Jakob used calculus to study the forms of many curves arising in practical situations, and studied mathematical probability (Ars conjectandi 1713); Bernoulli numbers are named for him. Johann developed exponential calculus and contributed to many areas of applied mathematics, including the problem of a particle moving in a gravitational field. His son, Daniel (1700-1782) worked on calculus and probability, and in physics proposed Bernoulli’s principle, which states that the pressure of a moving fluid decreases the faster it flows (which explains the origin of lift on the airfoil of an aircraft’s wing). This and other work on hydrodynamics was published in Hydrodynamica 1738.
perfectly round shape, the path of a point that moves so as to keep a constant distance from a fixed point (the center). Each circle has a radius (the distance from any point on the circle to the center), a circumference (the boundary of the circle), diameters (straight lines crossing the circle through the center), chords (lines joining two points on the circumference), tangents (lines that touch the circumference at one point only), sectors (regions inside the circle between two radii), and segments (regions between a chord and the circumference).
The ratio of the distance all around the circle (the circumference) to the diameter is an irrational number called p (pi), roughly equal to 3.1416. A circle of radius r and diameter d has a circumference C = pd, or C = 2pr, and an area A = pr2. The area of a circle can be shown by dividing it into very thin sectors and reassembling them to make an approximate rectangle. The proof of A = pr2 can be done only by using integral calculus.
branch of calculus involving applications such as the determination of maximum and minimum points and rates of change.
branch of mathematics using the process of integration. It is concerned with finding volumes and areas and summing infinitesimally small quantities.
in mathematics, a method in calculus of determining the solutions of definite or indefinite integrals.
An example of a definite integral can be thought of as finding the area under a curve (as represented by an algebraic expression or function) between particular values of the function's variable. In practice, integral calculus provides scientists with a powerful tool for doing calculations that involve a continually varying quantity (such as determining the position at any given instant of a space rocket that is accelerating away from Earth). Its basic principles were discovered in the late 1660s independently by the German philosopher Leibniz and the British scientist Newton.
pertaining to matrix or womb.
in mathematics, a square (n × n) or rectangular (m × n) array of elements (numbers or algebraic variables). They are a means of condensing information about mathematical systems and can be used for, among other things, solving simultaneous linear equations (see simultaneous equations and transformations.
Much early matrix theory was developed by the British mathematician Arthur Cayley, although the term was coined by his contemporary James Sylvester (1814-1897).
symbol used in counting or measuring. In mathematics, there are various kinds of numbers. The everyday number system is the decimal (“proceeding by tens”) system, using the base ten. Real numbers include all rational numbers (integers, or whole numbers, and fractions) and irrational numbers (those not expressible as fractions). Complex numbers include the real and unreal numbers (real-number multiples of the square root of -1). The binary number system, used in computers, has two as its base. The ordinary numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, give a counting system that, in the decimal system, continues 10, 11, 12, 13, and so on. These are whole numbers (integers), with fractions represented as, for example, 1/4 , 1/2 , 3/4 , or as decimal fractions (0.25, 0.5, 0.75). They are also rational numbers. Irrational numbers cannot be represented in this way and require symbols, such as Ö2, p, and e. They can be expressed numerically only as the (inexact) approximations 1.414, 3.142 and 2.718 (to three places of decimals) respectively. The symbols p and e are also examples of transcendental numbers, because they (unlike Ö2) cannot be derived by solving a polynomial equation (an equation with one variable quantity) with rational coefficients (multiplying factors). Complex numbers, which include the real numbers as well as unreal numbers, take the general form a + bi, where i = Ö-1 (that is, i2 = -1), and a is the real part and bi the unreal part.
history The ancient Egyptians, Greeks, Romans, and Babylonians all evolved number systems, although none had a zero, which was introduced from India by way of Arab mathematicians in about the 6th century AD and allowed a place-value system to be devised on which the decimal system is based. Other number systems have since evolved and have found applications. For example, numbers to base two (binary numbers), using only 0 and 1, are commonly used in digital computers to represent the two-state “on” or “off” pulses of electricity. Binary numbers were first developed by German mathematician Gottfried Leibniz in the late 17th century.
in mathematics, an algebraic expression that has one or more variables (denoted by letters). A polynomial of degree one, that is, whose highest power of x is 1, as in 2x + 1, is called a linear polynomial; 3x2 + 2x + 1 is quadratic; 4x3 + 3x2 + 2x + 1 is cubic.
in mathematics, a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x2). The general formula of such equations is ax2 + bx + c = 0, in which a, b, and c are real numbers, and only the coefficient a cannot equal 0.
In coordinate geometry, a quadratic function represents a parabola.
Some quadratic equations can be solved by factorization, or the values of x can be found by using the formula for the general solution x = [-b ± Ö(b2 -4ac)]/2a. Depending on the value of the discriminant b2 -4ac, a quadratic equation has two real, two equal, or two complex roots (solutions). When b2 -4ac > 0, there are two distinct real roots. When b2 - 4ac = 0, there are two equal real roots. When b2 - 4ac < 0, there are two distinct complex roots.