1.
Grade (slope)
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The grade of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of tilt, often slope is calculated as a ratio of rise to run, or as a fraction in which run is the horizontal distance and rise is the vertical distance. The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks, grades are typically specified for new linear constructions. The grade may refer to the slope or the perpendicular cross slope. There are several ways to express slope, as an angle of inclination to the horizontal, as a percentage, the formula for which is 100 rise run which could also be expressed as the tangent of the angle of inclination times 100. In the U. S. this percentage grade is the most commonly used unit for communicating slopes in transportation, surveying, construction, and civil engineering. As a per mille figure, the formula for which is 1000 rise run which could also be expressed as the tangent of the angle of inclination times 1000 and this is commonly used in Europe to denote the incline of a railway. As a ratio of one part rise to so many parts run, for example, a slope that has a rise of 5 feet for every 100 feet of run would have a slope ratio of 1 in 20. This is generally the method used to describe railway grades in Australia, any of these may be used. Grade is usually expressed as a percentage, but this is converted to the angle α from horizontal or the other expressions. Slope may still be expressed when the run is not known. This is not the way to specify slope, it follows the sine function rather than the tangent function. But in practice the way to calculate slope is to measure the distance along the slope and the vertical rise. When the angle of inclination is small, using the slope length rather than the horizontal displacement makes only an insignificant difference, Railway gradients are usually expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In any case, the identity holds for all inclinations up to 90 degrees, tan α = sin α1 − sin 2 α In Europe. Grades are related using the equations with symbols from the figure at top
2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
4.
Absolute value
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In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
5.
Constant function
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In mathematics, a constant function is a function whose value is the same for every input value. For example, the function y =4 is a constant function because the value of y is 4 regardless of the value x. As a real-valued function of an argument, a constant function has the general form y = c or just y = c. Example, The function y =2 or just y =2 is the constant function where the output value is c =2. The domain of function is the set of all real numbers ℝ. The codomain of this function is just, the independent variable x does not appear on the right side of the function expression and so its value is vacuously substituted. No matter what value of x is input, the output is 2, real-world example, A store where every item is sold for the price of 1 euro. The graph of the constant function y = c is a line in the plane that passes through the point. In the context of a polynomial in one variable x, the constant function is a polynomial of degree 0. This function has no point with the x-axis, that is. On the other hand, the polynomial f =0 is the zero function. It is the constant function and every x is a root and its graph is the x-axis in the plane. A constant function is a function, i. e. the graph of a constant function is symmetric with respect to the y-axis. In the context where it is defined, the derivative of a function is a measure of the rate of change of values with respect to change in input values. Because a constant function does not change, its derivative is 0 and this is often written, ′ =0. Namely, if y=0 for all numbers x, then y is a constant function. Example, Given the constant function y = −2, the derivative of y is the identically zero function y ′ = ′ =0. Every constant function whose domain and codomain are the same is idempotent, every constant function between topological spaces is continuous
6.
Gradient
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In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a variable, for functions of several variables. The gradient is a function, as opposed to a derivative. If f is a differentiable, real-valued function of several variables, like the derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, the components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. The Jacobian is the generalization of the gradient for vector-valued functions of several variables, a further generalization for a function between Banach spaces is the Fréchet derivative. Consider a room in which the temperature is given by a field, T. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, the magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point is H, the gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%, if, instead, the road goes around the hill at an angle, then it will have a shallower slope. This observation can be stated as follows. If the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when H is differentiable, the dot product of the gradient of H with a unit vector is equal to the directional derivative of H in the direction of that unit vector. The gradient of a function f is denoted ∇f or ∇→f where ∇ denotes the vector differential operator. The notation grad f is commonly used for the gradient. The gradient of f is defined as the vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is
7.
Geography
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Geography is a field of science devoted to the study of the lands, the features, the inhabitants, and the phenomena of Earth. The first person to use the word γεωγραφία was Eratosthenes, Geography is an all-encompassing discipline that seeks an understanding of the Earth and its human and natural complexities—not merely where objects are, but how they have changed and come to be. It is often defined in terms of the two branches of geography and physical geography. Geography has been called the world discipline and the bridge between the human and the physical sciences, Geography is a systematic study of the Earth and its features. Traditionally, geography has been associated with cartography and place names, although many geographers are trained in toponymy and cartology, this is not their main preoccupation. Geographers study the space and the temporal database distribution of phenomena, processes, because space and place affect a variety of topics, such as economics, health, climate, plants and animals, geography is highly interdisciplinary. The interdisciplinary nature of the approach depends on an attentiveness to the relationship between physical and human phenomena and its spatial patterns. Names of places. are not geography. know by heart a whole gazetteer full of them would not, in itself and this is a description of the world—that is Geography. In a word Geography is a Science—a thing not of mere names but of argument and reason, of cause, just as all phenomena exist in time and thus have a history, they also exist in space and have a geography. Geography as a discipline can be split broadly into two main fields, human geography and physical geography. The former largely focuses on the environment and how humans create, view, manage. The latter examines the environment, and how organisms, climate, soil, water. The difference between these led to a third field, environmental geography, which combines physical and human geography. Physical geography focuses on geography as an Earth science and it aims to understand the physical problems and the issues of lithosphere, hydrosphere, atmosphere, pedosphere, and global flora and fauna patterns. Physical geography can be divided into broad categories, including, Human geography is a branch of geography that focuses on the study of patterns. It encompasses the human, political, cultural, social, and it requires an understanding of the traditional aspects of physical and human geography, as well as the ways that human societies conceptualize the environment. Integrated geography has emerged as a bridge between the human and the geography, as a result of the increasing specialisation of the two sub-fields. Examples of areas of research in the environmental geography include, emergency management, environmental management, sustainability, geomatics is concerned with the application of computers to the traditional spatial techniques used in cartography and topography
8.
Civil engineering
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Civil engineering is traditionally broken into a number of sub-disciplines. It is the second-oldest engineering discipline after military engineering, and it is defined to distinguish non-military engineering from military engineering, Civil engineering takes place in the public sector from municipal through to national governments, and in the private sector from individual homeowners through to international companies. Engineering has been an aspect of life since the beginnings of human existence, during this time, transportation became increasingly important leading to the development of the wheel and sailing. The construction of pyramids in Egypt were some of the first instances of large structure constructions, the Romans developed civil structures throughout their empire, including especially aqueducts, insulae, harbors, bridges, dams and roads. In the 18th century, the civil engineering was coined to incorporate all things civilian as opposed to military engineering. The first self-proclaimed civil engineer was John Smeaton, who constructed the Eddystone Lighthouse, in 1771 Smeaton and some of his colleagues formed the Smeatonian Society of Civil Engineers, a group of leaders of the profession who met informally over dinner. Though there was evidence of some meetings, it was little more than a social society. In 1818 the Institution of Civil Engineers was founded in London, the institution received a Royal Charter in 1828, formally recognising civil engineering as a profession. The first private college to teach engineering in the United States was Norwich University. The first degree in engineering in the United States was awarded by Rensselaer Polytechnic Institute in 1835. The first such degree to be awarded to a woman was granted by Cornell University to Nora Stanton Blatch in 1905, throughout ancient and medieval history most architectural design and construction was carried out by artisans, such as stonemasons and carpenters, rising to the role of master builder. Knowledge was retained in guilds and seldom supplanted by advances, structures, roads and infrastructure that existed were repetitive, and increases in scale were incremental. Brahmagupta, an Indian mathematician, used arithmetic in the 7th century AD, based on Hindu-Arabic numerals, Civil engineers typically possess an academic degree in civil engineering. The length of study is three to five years, and the degree is designated as a bachelor of engineering. The curriculum generally includes classes in physics, mathematics, project management, design, after taking basic courses in most sub-disciplines of civil engineering, they move onto specialize in one or more sub-disciplines at advanced levels. In most countries, a degree in engineering represents the first step towards professional certification. After completing a degree program, the engineer must satisfy a range of requirements before being certified. Once certified, the engineer is designated as a engineer, a chartered engineer
9.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
