analytic geometry branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates , and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax + by + c =0 represents a straight line in the xy -plane, and the linear equation ax + by + cz + d =0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection.
In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x -axis. If the line is parallel to the x -axis, its slope is zero. Two or more lines with equal slopes are parallel to one another. In general, the slope of the line through the points ( x1 , y1 ) and ( x2 , y2 ) is given by m = ( y2 - y1 ) / ( x2 - x1 ). The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation ax2 + bxy + cy2 + dx + ey + f =0, where a, b, … , f are constants and a, b, and c are not all zero.
In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x-, y-, and z -axes, respectively; these satisfy the relationship λ 2 +μ 2 +ν 2 = 1. In the same way that the conic sections are studied in two dimensions, the 17 quadric surfaces, e.g., the ellipsoid, paraboloid, and elliptic paraboloid, are studied in solid analytic geometry in terms of the general equation ax2 + by2 + cz2 + dxy + exz + fyz + px + qy + rz + s =0.
The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry. Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry .
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Cartesian coordinates
| By Calculus for All
Cartesian coordinates [for René Descartes ], system for representing the relative positions of points in a plane or in space. In a plane, the point P is specified by the pair of numbers ( x,y ) representing the distances of the point from two intersecting straight lines, referred to as the x -axis and the y -axis. The point of intersection of these axes, which are called the coordinate axes, is known as the origin. In rectangular coordinates, the type most often used, the axes are taken to be perpendicular, with the x -axis horizontal and the y -axis vertical, so that the x -coordinate, or abscissa, of P is measured along the horizontal perpendicular from P to the y -axis (i.e., parallel to the x -axis) and the y -coordinate, or ordinate, is measured along the vertical perpendicular from P to the x -axis (parallel to the y -axis). In oblique coordinates the axes are not perpendicular; the abscissa of P is measured along a parallel to the x -axis, and the ordinate is measured along a parallel to the y -axis, but neither of these parallels is perpendicular to the other coordinate axis as in rectangular coordinates. Similarly, a point in space may be specified by the triple of numbers ( x,y,z ) representing the distances from three planes determined by three intersecting straight lines not all in the same plane; i.e., the x -coordinate represents the distance from the yz -plane measured along a parallel to the x -axis, the y -coordinate represents the distance from the xz -plane measured along a parallel to the y -axis, and the z -coordinate represents the distance from the xy -plane measured along a parallel to the z -axis (the axes are usually taken to be mutually perpendicular). Analogous systems may be defined for describing points in abstract spaces of four or more dimensions. Many of the curves studied in classical geometry can be described as the set of points ( x,y ) that satisfy some equation f(x,y) =0. In this way certain questions in geometry can be transformed into questions about numbers and resolved by means of analytic geometry .
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Taken From : http://www.encyclopedia.com/topic/Cartesian_coordinates.aspx#1E1-Cartes-coo
Extension of the Trigonometric Functions
| By Calculus for All
The notion of the trigonometric functions can be extended beyond 90° by defining the functions with respect to Cartesian coordinates . Let r be a line of unit length from the origin to the point P ( x,y ), and let θ be the angle r makes with the positive x -axis. The six functions become sin θ = y / r = y, cos θ= x / r = x, tan θ= y / x, cot θ= x / y, sec θ= r / x =1/ x, and csc θ= r / y =1/ y. As θ increases beyond 90°, the point P crosses the y -axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270° P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive. Since the positions of r for angles of 360° or more coincide with those already taken by r as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on. This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.
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Taken From : http://www.encyclopedia.com/topic/trigonometry.aspx#1E1-trigonom
The Basic Trigonometric Functions
| By Calculus for All
Trigomometry originated as the study of certain mathematical relations originally defined in terms of the angles and sides of a right triangle, i.e., one containing a right angle (90°). Six basic relations, or trigonometric functions, are defined. If A, B, and C are the measures of the angles of a right triangle ( C =90°) and a, b, and c are the lengths of the respective sides opposite these angles, then the six functions are expressed for one of the acute angles, say A, as various ratios of the opposite side ( a ), the adjacent side ( b ), and the hypotenuse ( c ), as set out in the table. Although the actual lengths of the sides of a right triangle may have any values, the ratios of the lengths will be the same for all similar right triangles, large or small; these ratios depend only on the angles and not on the actual lengths. The functions occur in pairs—sine and cosine, tangent and cotangent, secant and cosecant—called cofunctions. In equations they are usually represented as sin, cos, tan, cot, sec, and csc. Since in ordinary (Euclidean) plane geometry the sum of the angles of a triangle is 180°, angles A and B must add up to 90° and therefore are complementary angles. From the definitions of the functions, it may be seen that sin B =cos A, cos B =sin A, tan B =cot A, and sec B =csc A ; in general, the function of an angle is equal to the cofunction of its complement. Since the hypotenuse ( c ), is always the longest side of a right triangle, the values of the sine and cosine are always between zero and one, the values of the secant and cosecant are always equal to or greater than one, and the values of the tangent and cotangent are unbounded, increasing from zero without limit.
For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 : 3 : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°= 3 /2, tan 30°=cot 60°=1/ 3 , cot 30°=tan 60°= 3 , sec 30°=csc 60°=2/ 3 , and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.
A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a /sin A = b /sin B = c /sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a2 = b2 + c2 -2 bc cos A and the Law of Tangents holds that ( a - b )/( a + b )=[tan 1/2 ( A - B )]/[tan 1/2 ( A + B )]. Each of the trigonometric functions can be represented by an infinite series .
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For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 : 3 : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°= 3 /2, tan 30°=cot 60°=1/ 3 , cot 30°=tan 60°= 3 , sec 30°=csc 60°=2/ 3 , and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.
A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a /sin A = b /sin B = c /sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a2 = b2 + c2 -2 bc cos A and the Law of Tangents holds that ( a - b )/( a + b )=[tan 1/2 ( A - B )]/[tan 1/2 ( A + B )]. Each of the trigonometric functions can be represented by an infinite series .
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The Differential Calculus
| By Calculus for All
The differential calculus arises from the study of the limit of a quotient, Δ y /Δ x, as the denominator Δ x approaches zero, where x and y are variables. y may be expressed as some function of x, or f(x), and Δ y and Δ x represent corresponding increments, or changes, in y and x. The limit of Δ y /Δ x is called the derivative of y with respect to x and is indicated by dy / dx or Dxy : The symbols dy and dx are called differentials (they are single symbols, not products), and the process of finding the derivative of y = f(x) is called differentiation. The derivative dy / dx = df(x) / dx is also denoted by y′, or f′(x). The derivative f′(x) is itself a function of x and may be differentiated, the result being termed the second derivative of y with respect to x and denoted by y″, f″(x), or d2y / dx2 . This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if y = xn , then y′ =nx n - 1 , and if y =sin x, then y′ =cos x (see trigonometry ). In general, the derivative of y with respect to x expresses the rate of change in y for a change in x. In physical applications the independent variable (here x ) is frequently time; e.g., if s = f ( t ) expresses the relationship between distance traveled, s, and time elapsed, t, then s′ = f′ ( t ) represents the rate of change of distance with time, i.e., the speed, or velocity.
Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f′ ( t )= ds / dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y = f ( x ) is a real-valued function of a real variable, the ratio Δ y /Δ x =( y2 - y1 )/( x2 - x1 ) represents the slope of a straight line through the two points P ( x1 , y1 ) and Q ( x2 , y2 ) on the graph of the function. If P is taken closer to Q, then x1 will approach x2 and Δ x will approach zero. In the limit where Δ x approaches zero, the ratio becomes the derivative dy / dx = f′ ( x ) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.
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Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f′ ( t )= ds / dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y = f ( x ) is a real-valued function of a real variable, the ratio Δ y /Δ x =( y2 - y1 )/( x2 - x1 ) represents the slope of a straight line through the two points P ( x1 , y1 ) and Q ( x2 , y2 ) on the graph of the function. If P is taken closer to Q, then x1 will approach x2 and Δ x will approach zero. In the limit where Δ x approaches zero, the ratio becomes the derivative dy / dx = f′ ( x ) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.
Taken From : http://www.encyclopedia.com/topic/calculus.aspx
The Integral Calculus
| By Calculus for All
The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve y = f ( x ) between two values of x, say a and b. Let the interval between a and b be divided into n subintervals, from a = x0 through x1 , x2 , x3 , … xi - 1 , xi , … , up to xn = b. The width of a given subinterval is equal to the difference between the adjacent values of x, or Δ xi = xi - xi - 1 , where i designates the typical, or i th, subinterval. On each Δ xi a rectangle can be formed of width Δ xi , height yi = f ( xi ) (the value of the function corresponding to the value of x on the right-hand side of the subinterval), and area Δ Ai = f ( xi )Δ xi . In some cases, the rectangle may extend above the curve, while in other cases it may fail to include some of the area under the curve; however, if the areas of all these rectangles are added together, the sum will be an approximation of the area under the curve.
This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the Δ x 's and the amounts by which the Δ A 's exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest Δ x approaches zero), the sum is equal to the area under the curve: The last expression on the right is called the integral of f ( x ), and f ( x ) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.
An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F ( x ) is a function whose derivative is f ( x ), then the area under the graph of y = f ( x ) between a and b is equal to F ( b ) - F ( a ). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols: The function F ( x ), which is equal to the integral of f ( x ), is sometimes called an antiderivative of f ( x ), while the process of finding F ( x ) from f ( x ) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F ( x ) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f ( x ) is The value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z = f ( x,y ) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂ z /∂ x and ∂ z /∂ y or by Dxz and Dyz. Three different second derivatives are possible, ∂ 2z /∂ x2 , ∂ 2z /∂ y2 , and ∂ 2z /∂ x ∂ y =∂ 2z /∂ y ∂ x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.
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This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the Δ x 's and the amounts by which the Δ A 's exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest Δ x approaches zero), the sum is equal to the area under the curve: The last expression on the right is called the integral of f ( x ), and f ( x ) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.
An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F ( x ) is a function whose derivative is f ( x ), then the area under the graph of y = f ( x ) between a and b is equal to F ( b ) - F ( a ). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols: The function F ( x ), which is equal to the integral of f ( x ), is sometimes called an antiderivative of f ( x ), while the process of finding F ( x ) from f ( x ) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F ( x ) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f ( x ) is The value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z = f ( x,y ) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂ z /∂ x and ∂ z /∂ y or by Dxz and Dyz. Three different second derivatives are possible, ∂ 2z /∂ x2 , ∂ 2z /∂ y2 , and ∂ 2z /∂ x ∂ y =∂ 2z /∂ y ∂ x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.
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