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What did Fermat contribute to the field of calculus?
Fermat contributed to the development of calculus. His study of curves and equations prompted him to generalize the equation for the ordinary parabola ay=x2, and that for the rectangular hyperbola xy=a2, to the form an-1y=xn. The curves determined by this equation are known as the parabolas or hyperbolas of Fermat according as n is positive or negative (Kolata). He similarly generalized the Archimedean spiral, r=aQ. In the 1630s, these curves then directed him to an algorithm, or rule of mathematical procedure, that was equivalent to differentiation. This procedure enabled hi m to find tangents to curves and locate maximum, minimum, and inflection points of polynomials (Kolata). His main contribution was finding the tangents of a curve as well as its points of extrema. He believed that his tangent-finding method was an extension of his method for locating extrema (Rosenthal, page 79). For any equation, Fermat 's method for finding the tangent at a given point actually finds the subtangent for that specific point (Eves, page 326). Fermat found the areas bounded by these curves through a summation process. "The creators of calculus, including Fermat, reli ed on geometric and physical (mostly kinematical and dynamical) intuition to get them ahead: they looked at what passed in their imaginations for the graph of a continuous curve..." (Bell, page 59). This process is now called integral calculus. Fermat founded formulas for areas bounded by these curves through a summation process that is now used for the same purpose in integral calculus. Such a formula is: A= xndx = an+1 / (n + 1) It is not known whether or not Fermat noticed that differentiation of xn, leading to nan-1, is the inverse of integrating xn. Through skillful transformations, he handled problems involving more general algebraic curves. Fermat applied his analysis of infinitesimal quantities to a variety of other problems, including the calculation of centers of gravity and finding the length of curves (Mahoney, pages 47, 156, 204-205). Fermat was unable to notice what is now considered the Fundamental Theorem of Calculus, however, his work on this subject aided in the development of differential calculus (Parker, page 304). Additionally, he contributed to the law of refraction by disagreeing with his contemporary, the philosopher and amateur mathematician, René Descartes. Fermat claimed that Descartes had incorrectly deduced his law of refraction since it was deep-seated in his assumptions. As a result, Desc artes was irritated and attacked Fermat's method of maxima, minima, and tangents (Mahoney, pages 170-195). Fermat differed with Cartesian views concerning the law of refraction, published by Descartes in 1637 in La Dioptrique. Descartes attempted to justify the sine law through an assumption that light travels more rapidly in the denser of the two media involved in the refraction. (Mahoney, page 65). Twenty years later, Fermat noted th at this appeared to be in conflict with the view of the Aristotelians that nature always chooses the shortest path. "According to [Fermat's] principle, if a ray of light passes from a point A to another point B, being reflected and refracted in any manner during the passage, the path which it must take can be calculated...th e time taken to pass from A to B shall be an extreme" (Bell, page 63). Applying his method of maxima and minima, Fermat made the assumption that light travels less rapidly in the denser medium and showed that the law of refraction is concordant with his "principle of least time." "From this principle, Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence (in refraction) is a constant number times the sine of the angle of refraction in passing from one medium to anot her" (Bell, page 63). His argument concerning the speed of light was found later to be in agreement with the wave theory of the 17th-century Dutch scientist Huygens, and was verified experimentally in 1869 by Fizeau. In addition to the law of refraction, Fermat obtained the subtangent to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e. There is nothing to indicate that he was aware that the process was general, and it is likely that he never separated it his method from the context of the particular problems he was considering (Coolidge, page 458). The first definite statement of the method was due to Barrow, and was published in 1669. Fermat also obtained the areas of parabolas and hyperbolas of any order, and determined the centers of mass of a few simple laminae and of a paraboloid of revolution (Ball, pages 49, 77 , 108). Fermat was also strongly influenced by Viète, who revived interest in Greek analysis. The ancient Greeks divided their geometric arguments into two categories: analysis and synthesis. While analysis meant "assuming the pro position in question and deducing from it something already known," synthesis is what we now call "proof" (Mahoney, page 30). Fermat recognized the need for synthesis, but he would often give an analysis of a theorem. He would then state that it could easily be converted to a synthesis. Source:http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html
What is the law of contrapositive?
If p->q, then the law of the contrapositive is that not q -> not p
What is the answer for 5c equals -75?
The 'answer' is the number that 'c' must be, if 5c is really the same as -75.In order to find out what number that is, you could use 'algebra'.First, write the equation, so that you can look at it:5c = -75Now, use the law of algebra that says: "If equals are divided by equals,then the quotients are equal".The left and right sides of your equation are equals. Divide them both by 5,and that law says that the quotients on both sides will be equal:c = -15
What is the purpose behind Navier Strokes equation?
The Navier-Strokes equation is a term in physics used to describe the motion of a fluid substance. The equation applies Newton's second law to fluid motion.
What is a Derived adverb?
A derivative adverb would be the adverb form of a derivative adjective: one formed from a noun or a verb by the addition of a suffix (and possibly by a prefix as well). An example would be the adjective lawful (from the noun law) and the adverb lawfully.
How would you predict weather an angle of refraction is larger or smaller than the angle of incidence?
use snells law
Who discovered law of refraction of light?
snells
The bending of a wave when it moves from one material into another?
It is Refraction
What are the snells law of refraction?
Snell's Law of Refraction describes how light waves bend when they pass from one medium to another with a different refractive index. It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the velocities of light in the two media. This law helps determine how much a light ray will bend as it moves from one medium to another.
The bending of a wave as it moves from one medium to another.?
refraction
What do the following terms mean-Snells law-Threshold frequency-instataneous velocity-wave mechanics-Youngs modulus-zeean effect?
Snell's law is a description of the relationship between the angles of incidence and refraction. Instantaneous Velocity is the velocity at one point.
How do you measure the angle of refracted light wave?
The angle of a refracted light wave is typically measured with respect to the normal, which is an imaginary line perpendicular to the boundary between two mediums. This angle is known as the angle of refraction and can be determined using Snell's Law, which relates the angles of incidence and refraction to the refractive indices of the two mediums.
The law of reflection states that the angle of incidence is equal to the angle of refraction?
The law of reflection states that the angle of incidence is equal to the angle of REFLECTION. Refraction is an entirely different phenomenon and has its own law.
The relationship between the angle of incidence and the angle of refraction is known as law?
Snell's Law describes the relationship between the angle of incidence and the angle of refraction for light passing through different mediums. It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media.
How does the angle of incidence compared with the angle of refraction for sound?
For refraction, the general relationship is given by Snell's Law.
What is the bending of waves as it passes at an angle from one medium to another?
The bending of waves as they pass at an angle from one medium to another is called refraction. This phenomenon occurs due to the change in speed of the wave as it travels through media with different densities, causing it to change direction. Refraction is governed by Snell's Law, which relates the angle of incidence to the angle of refraction.
What is snells law of refraction?
Snell's Law of Refraction describes how light bends when it passes from one medium to another with a different optical density. It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the velocities of light in the two mediums. Mathematically, it can be written as n1 sinθ1 = n2 sinθ2, where n1 and n2 are the refractive indices of the two mediums.
