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58-643: Transportation engineering or transport engineering is the application of technology and scientific principles to the planning, functional design, operation and management of facilities for any mode of transportation to provide for the safe, efficient, rapid, comfortable, convenient, economical, and environmentally compatible movement of people and goods transport. The planning aspects of transportation engineering relate to elements of urban planning , and involve technical forecasting decisions and political factors. Technical forecasting of passenger travel usually involves an urban transportation planning model , requiring

116-695: A 2 ≥ 0 {\displaystyle a^{2}\geq 0} is true for all real numbers a , and is therefore a law. Laws over an equality are called identities . For example, ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} and cos 2 ⁡ θ + sin 2 ⁡ θ = 1 {\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1} are identities. Mathematical laws are distinguished from scientific laws which are based on observations , and try to describe or predict

174-579: A physical system under repeated conditions, and it implies that there is a causal relationship involving the elements of the system. Factual and well-confirmed statements like "Mercury is liquid at standard temperature and pressure" are considered too specific to qualify as scientific laws. A central problem in the philosophy of science , going back to David Hume , is that of distinguishing causal relationships (such as those implied by laws) from principles that arise due to constant conjunction . Laws differ from scientific theories in that they do not posit

232-463: A 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation . The rules can be expressed in English as: The three Laws of thought are: Benford's law is an observation that in many real-life sets of numerical data , the leading digit is likely to be small. In sets that obey the law, the number 1 appears as

290-420: A law or theory from facts. Calling a law a fact is ambiguous , an overstatement , or an equivocation . The nature of scientific laws has been much discussed in philosophy , but in essence scientific laws are simply empirical conclusions reached by scientific method; they are intended to be neither laden with ontological commitments nor statements of logical absolutes . A scientific law always applies to

348-476: A mechanism or explanation of phenomena: they are merely distillations of the results of repeated observation. As such, the applicability of a law is limited to circumstances resembling those already observed, and the law may be found to be false when extrapolated. Ohm's law only applies to linear networks; Newton's law of universal gravitation only applies in weak gravitational fields; the early laws of aerodynamics , such as Bernoulli's principle , do not apply in

406-426: A particular phenomenon always occurs if certain conditions be present". The production of a summary description of our environment in the form of such laws is a fundamental aim of science . Several general properties of scientific laws, particularly when referring to laws in physics , have been identified. Scientific laws are: The term "scientific law" is traditionally associated with the natural sciences , though

464-430: A point; hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region (see the main article for details). In the table below, the fluxes flows for various physical quantities in transport, and their associated continuity equations, are collected for comparison. u = velocity field of fluid (m s ) Ψ = wavefunction of quantum system More general equations are

522-441: A range of natural phenomena . The more significant laws are often called theorems . Triangle inequality : If a , b , and c are the lengths of the sides of a triangle then the triangle inequality states that with equality only in the degenerate case of a triangle with zero area . In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths ( norms ): where

580-403: A series of improving and more precise generalizations. Scientific laws are typically conclusions based on repeated scientific experiments and observations over many years and which have become accepted universally within the scientific community . A scientific law is " inferred from particular facts, applicable to a defined group or class of phenomena , and expressible by the statement that

638-432: A triangle. Only the former are covered in this article. These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the integration of non-trigonometric functions: a common technique which involves first using the substitution rule with a trigonometric function , and then simplifying the resulting integral with a trigonometric identity. One of

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696-418: Is a combination of extensive evidence of something not occurring, combined with an underlying theory , very successful in making predictions, whose assumptions lead logically to the conclusion that something is impossible. While an impossibility assertion in natural science can never be absolutely proved, it could be refuted by the observation of a single counterexample . Such a counterexample would require that

754-467: Is a consequence of the shift symmetry of time (no moment of time is different from any other), while conservation of momentum is a consequence of the symmetry (homogeneity) of space (no place in space is special, or different from any other). The indistinguishability of all particles of each fundamental type (say, electrons, or photons) results in the Dirac and Bose quantum statistics which in turn result in

812-705: Is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm: Moreover, the two sides are equal if and only if u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } are linearly dependent . Pigeonhole principle : If n items are put into m containers, with n > m , then at least one container must contain more than one item. For example, of three gloves (none of which

870-441: Is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into. De Morgan's laws : In propositional logic and Boolean algebra , De Morgan's laws , also known as De Morgan's theorem , are a pair of transformation rules that are both valid rules of inference . They are named after Augustus De Morgan ,

928-505: Is one of the main goals of science. The fact that laws have never been observed to be violated does not preclude testing them at increased accuracy or in new kinds of conditions to confirm whether they continue to hold, or whether they break, and what can be discovered in the process. It is always possible for laws to be invalidated or proven to have limitations, by repeatable experimental evidence, should any be observed. Well-established laws have indeed been invalidated in some special cases, but

986-413: Is only true for certain values of θ {\displaystyle \theta } , not all. For example, this equation is true when θ = 0 , {\displaystyle \theta =0,} but false when θ = 2 {\displaystyle \theta =2} . Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g.

1044-544: Is the inner product . Examples of inner products include the real and complex dot product ; see the examples in inner product . Every inner product gives rise to a Euclidean l 2 {\displaystyle l_{2}} norm , called the canonical or induced norm , where the norm of a vector u {\displaystyle \mathbf {u} } is denoted and defined by where ⟨ u , u ⟩ {\displaystyle \langle \mathbf {u} ,\mathbf {u} \rangle }

1102-425: Is the mathematical consequence of the 3-dimensionality of space . One strategy in the search for the most fundamental laws of nature is to search for the most general mathematical symmetry group that can be applied to the fundamental interactions. Conservation laws are fundamental laws that follow from the homogeneity of space, time and phase , in other words symmetry . Conservation laws can be expressed using

1160-510: The Pauli exclusion principle for fermions and in Bose–Einstein condensation for bosons . Special relativity uses rapidity to express motion according to the symmetries of hyperbolic rotation , a transformation mixing space and time. Symmetry between inertial and gravitational mass results in general relativity . The inverse square law of interactions mediated by massless bosons

1218-459: The convection–diffusion equation and Boltzmann transport equation , which have their roots in the continuity equation. Classical mechanics, including Newton's laws , Lagrange's equations , Hamilton's equations , etc., can be derived from the following principle: where S {\displaystyle {\mathcal {S}}} is the action ; the integral of the Lagrangian of

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1276-565: The hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a , b and the hypotenuse c , sometimes called the Pythagorean equation: Geometrically, trigonometric identities are identities involving certain functions of one or more angles . They are distinct from triangle identities , which are identities involving both angles and side lengths of

1334-545: The inner product between two vectors in an inner product space in terms of the product of the vector norms . It is considered one of the most important and widely used inequalities in mathematics. The Cauchy–Schwarz inequality states that for all vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } of an inner product space where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle }

1392-950: The law of conservation of energy can be written as Δ E = 0 {\displaystyle \Delta E=0} , where E {\displaystyle E} is the total amount of energy in the universe. Similarly, the first law of thermodynamics can be written as d U = δ Q − δ W {\displaystyle \mathrm {d} U=\delta Q-\delta W\,} , and Newton's second law can be written as F = d p d t . {\displaystyle \textstyle F={\frac {dp}{dt}}.} While these scientific laws explain what our senses perceive, they are still empirical (acquired by observation or scientific experiment) and so are not like mathematical theorems which can be proved purely by mathematics. Like theories and hypotheses, laws make predictions; specifically, they predict that new observations will conform to

1450-453: The social sciences also contain laws. For example, Zipf's law is a law in the social sciences which is based on mathematical statistics . In these cases, laws may describe general trends or expected behaviors rather than being absolutes. In natural science, impossibility assertions come to be widely accepted as overwhelmingly probable rather than considered proved to the point of being unchallengeable. The basis for this strong acceptance

1508-474: The (more famous) mass–energy equivalence E = mc is a special case. General relativity is governed by the Einstein field equations , which describe the curvature of space-time due to mass–energy equivalent to the gravitational field. Solving the equation for the geometry of space warped due to the mass distribution gives the metric tensor . Using the geodesic equation, the motion of masses falling along

1566-533: The 2nd, zero resultant acceleration): where p = momentum of body, F ij = force on body i by body j , F ji = force on body j by body i . For a dynamical system the two equations (effectively) combine into one: in which F E = resultant external force (due to any agent not part of system). Body i does not exert a force on itself. From the above, any equation of motion in classical mechanics can be derived. Equations describing fluid flow in various situations can be derived, using

1624-495: The Galilean transformations for low velocities much less than the speed of light c . The magnitudes of 4-vectors are invariants – not "conserved", but the same for all inertial frames (i.e. every observer in an inertial frame will agree on the same value), in particular if A is the four-momentum , the magnitude can derive the famous invariant equation for mass–energy and momentum conservation (see invariant mass ): in which

1682-480: The Lagrangian, is required (in other words it is not as simple as "differentiating a function and setting it to zero, then solving the equations to find the points of maxima and minima etc", rather this idea is applied to the entire "shape" of the function, see calculus of variations for more details on this procedure). Notice L is not the total energy E of the system due to the difference, rather than

1740-581: The Lorentz transformation). Similarly, the Newtonian gravitation law is a low-mass approximation of general relativity, and Coulomb's law is an approximation to quantum electrodynamics at large distances (compared to the range of weak interactions). In such cases it is common to use the simpler, approximate versions of the laws, instead of the more accurate general laws. Laws are constantly being tested experimentally to increasing degrees of precision, which

1798-664: The above classical equations of motion and often conservation of mass, energy and momentum. Some elementary examples follow. Some of the more famous laws of nature are found in Isaac Newton 's theories of (now) classical mechanics , presented in his Philosophiae Naturalis Principia Mathematica , and in Albert Einstein 's theory of relativity . The two postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of relative motion . They can be stated as "the laws of physics are

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1856-436: The assumptions underlying the theory that implied the impossibility be re-examined. Some examples of widely accepted impossibilities in physics are perpetual motion machines , which violate the law of conservation of energy , exceeding the speed of light , which violates the implications of special relativity , the uncertainty principle of quantum mechanics , which asserts the impossibility of simultaneously knowing both

1914-399: The case of compressible flow such as occurs in transonic and supersonic flight; Hooke's law only applies to strain below the elastic limit ; Boyle's law applies with perfect accuracy only to the ideal gas, etc. These laws remain useful, but only under the specified conditions where they apply. Many laws take mathematical forms, and thus can be stated as an equation; for example,

1972-412: The definition of generalized momentum, there is the symmetry: The Hamiltonian as a function of generalized coordinates and momenta has the general form: Newton's laws of motion They are low-limit solutions to relativity . Alternative formulations of Newtonian mechanics are Lagrangian and Hamiltonian mechanics. The laws can be summarized by two equations (since the 1st is a special case of

2030-575: The design, construction, and operation of railroads and mass transit systems that use a fixed guideway (such as light rail or monorails ). Typical tasks include: Railway engineers work to build a cleaner and safer transportation network by reinvesting and revitalizing the rail system to meet future demands. In the United States, railway engineers work with elected officials in Washington, D.C., on rail transportation issues to make sure that

2088-499: The double-angle identity sin ⁡ ( 2 θ ) = 2 sin ⁡ θ cos ⁡ θ {\displaystyle \sin(2\theta )=2\sin \theta \cos \theta } , the addition formula for tan ⁡ ( x + y ) {\displaystyle \tan(x+y)} ), which can be used to break down expressions of larger angles into those with smaller constituents. Cauchy–Schwarz inequality : An upper bound on

2146-404: The dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the generalized coordinates in the configuration space , i.e. the curve q ( t ), parameterized by time (see also parametric equation for this concept). The action is a functional rather than a function , since it depends on the Lagrangian, and the Lagrangian depends on

2204-771: The engineer create business models to complete accurate forecasts of the future conditions of the system. Operations and management involve traffic engineering , so that vehicles move smoothly on the road or track. Older techniques include signs , signals , markings , and tolling . Newer technologies involve intelligent transportation systems , including advanced traveler information systems (such as variable message signs ), advanced traffic control systems (such as ramp meters ), and vehicle infrastructure integration . Human factors are an aspect of transportation engineering, particularly concerning driver-vehicle interface and user interface of road signs, signals, and markings. Engineers in this specialization: Railway engineers handle

2262-450: The estimation of trip generation , trip distribution , mode choice , and route assignment . More sophisticated forecasting can include other aspects of traveler decisions, including auto ownership, trip chaining (the decision to link individual trips together in a tour) and the choice of residential or business location (known as land use forecasting ). Passenger trips are the focus of transportation engineering because they often represent

2320-590: The facility has), determining the materials and thickness used in pavement designing the geometry (vertical and horizontal alignment) of the roadway (or track). Before any planning occurs an engineer must take what is known as an inventory of the area or, if it is appropriate, the previous system in place. This inventory or database must include information on population, land use, economic activity, transportation facilities and services, travel patterns and volumes, laws and ordinances, regional financial resources, and community values and expectations. These inventories help

2378-400: The general continuity equation (for a conserved quantity) can be written in differential form as: where ρ is some quantity per unit volume, J is the flux of that quantity (change in quantity per unit time per unit area). Intuitively, the divergence (denoted ∇⋅) of a vector field is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at

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2436-396: The geodesics can be calculated. Law (mathematics) In mathematics , a law is a formula that is always true within a given context. Laws describe a relationship , between two or more expressions or terms (which may contain variables ), usually using equality or inequality , or between formulas themselves, for instance, in mathematical logic . For example, the formula

2494-416: The given law. Laws can be falsified if they are found in contradiction with new data. Some laws are only approximations of other more general laws, and are good approximations with a restricted domain of applicability. For example, Newtonian dynamics (which is based on Galilean transformations) is the low-speed limit of special relativity (since the Galilean transformation is the low-speed approximation to

2552-431: The impacts and demands of aircraft in their design of airport facilities. These engineers must use the analysis of predominant wind direction to determine runway orientation, determine the size of runway border and safety areas, different wing tip to wing tip clearances for all gates and must designate the clear zones in the entire port. The Civil Engineering Department, consisting of Civil and Structural Engineers, undertakes

2610-401: The length of the third side has been replaced by the length of the vector sum u + v . When u and v are real numbers, they can be viewed as vectors in R 1 {\displaystyle \mathbb {R} ^{1}} , and the triangle inequality expresses a relationship between absolute values . Pythagorean theorem : It states that the area of the square whose side is

2668-397: The most prominent examples of trigonometric identities involves the equation sin 2 ⁡ θ + cos 2 ⁡ θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,} which is true for all real values of θ {\displaystyle \theta } . On the other hand, the equation

2726-447: The new formulations created to explain the discrepancies generalize upon, rather than overthrow, the originals. That is, the invalidated laws have been found to be only close approximations, to which other terms or factors must be added to cover previously unaccounted-for conditions, e.g. very large or very small scales of time or space, enormous speeds or masses, etc. Thus, rather than unchanging knowledge, physical laws are better viewed as

2784-451: The outcome of an experiment. Laws differ from hypotheses and postulates , which are proposed during the scientific process before and during validation by experiment and observation. Hypotheses and postulates are not laws, since they have not been verified to the same degree, although they may lead to the formulation of laws. Laws are narrower in scope than scientific theories , which may entail one or several laws. Science distinguishes

2842-406: The path q ( t ), so the action depends on the entire "shape" of the path for all times (in the time interval from t 1 to t 2 ). Between two instants of time, there are infinitely many paths, but one for which the action is stationary (to the first order) is the true path. The stationary value for the entire continuum of Lagrangian values corresponding to some path, not just one value of

2900-491: The peak of demand on any transportation system. A review of descriptions of the scope of various committees indicates that while facility planning and design continue to be the core of the transportation engineering field, such areas as operations planning, logistics, network analysis, financing, and policy analysis are also important, particularly to those working in highway and urban transportation. The National Council of Examiners for Engineering and Surveying (NCEES) list online

2958-565: The physical system between two times t 1 and t 2 . The kinetic energy of the system is T (a function of the rate of change of the configuration of the system), and potential energy is V (a function of the configuration and its rate of change). The configuration of a system which has N degrees of freedom is defined by generalized coordinates q = ( q 1 , q 2 , ... q N ). There are generalized momenta conjugate to these coordinates, p = ( p 1 , p 2 , ..., p N ), where: The action and Lagrangian both contain

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3016-692: The position and the momentum of a particle, and Bell's theorem : no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. Some laws reflect mathematical symmetries found in nature (e.g. the Pauli exclusion principle reflects identity of electrons, conservation laws reflect homogeneity of space , time, and Lorentz transformations reflect rotational symmetry of spacetime ). Many fundamental physical laws are mathematical consequences of various symmetries of space, time, or other aspects of nature. Specifically, Noether's theorem connects some conservation laws to certain symmetries. For example, conservation of energy

3074-401: The rail system meets the country's transportation needs. Railroad engineers can also move into the specialized field of train dispatching which focuses on train movement control. Port and harbor engineers handle the design, construction, and operation of ports, harbors, canals, and other maritime facilities. Airport engineers design and construct airports. Airport engineers must account for

3132-629: The results of experiments or observations, usually within a certain range of application. In general, the accuracy of a law does not change when a new theory of the relevant phenomenon is worked out, but rather the scope of the law's application, since the mathematics or statement representing the law does not change. As with other kinds of scientific knowledge, scientific laws do not express absolute certainty, as mathematical laws do. A scientific law may be contradicted, restricted, or extended by future observations. A law can often be formulated as one or several statements or equations , so that it can predict

3190-438: The safety protocols, geometric design requirements, and signal timing. Transportation engineering, primarily involves planning, design, construction, maintenance, and operation of transportation facilities. The facilities support air, highway, railroad, pipeline, water, and even space transportation. The design aspects of transportation engineering include the sizing of transportation facilities (how many lanes or how much capacity

3248-512: The same in all inertial frames " and "the speed of light is constant and has the same value in all inertial frames". The said postulates lead to the Lorentz transformations – the transformation law between two frame of references moving relative to each other. For any 4-vector this replaces the Galilean transformation law from classical mechanics. The Lorentz transformations reduce to

3306-459: The structural design of passenger, terminal design and cargo terminals, aircraft hangars (for parking commercial, private and government aircraft), runways and other pavements, technical buildings for installation of airport ground aids etc. for the airports in-house requirements and consultancy projects. They are even responsible for the master plan for airports they are authorized to work with. Scientific principle Scientific laws summarize

3364-554: The sum: The following general approaches to classical mechanics are summarized below in the order of establishment. They are equivalent formulations. Newton's is commonly used due to simplicity, but Hamilton's and Lagrange's equations are more general, and their range can extend into other branches of physics with suitable modifications. S = ∫ t 1 t 2 L d t {\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t\,\!} Using

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