Simpson's rules are a set of rules used in ship stability and naval architecture, to calculate the areas and volumes of irregular figures. [1] This is an application of Simpson's rule for finding the values of an integral, here interpreted as the area under a curve.
Also known as the 1–4–1 rule (after the multipliers used ). [2]
Also known as the 1–3–3–1 rule, Simpson's second rule is a simplified version of Simpson's 3/8 rule. [3]
Also known as the 5–8–1 rule, [4] SImpson's third rule is used to find the area between two consecutive ordinates when three consecutive ordinates are known. [5]
This estimates the area in the left half of the figure for Simpson's 1st Rule while using all three pieces of data.
Simpson's rules are used to calculate the volume of lifeboats, [6] and by surveyors to calculate the volume of sludge in a ship's oil tanks. For instance, in the latter, Simpson's 3rd rule is used to find the volume between two co-ordinates. To calculate the entire area / volume, Simpson's first rule is used. [7]
Simpson's rules are used by a ship's officers to check that the area under the ship's GZ curve complies with IMO stability criteria.
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, ..., an, the geometric mean is defined as
A hull is the watertight body of a ship, boat, submarine, or flying boat. The hull may open at the top, or it may be fully or partially covered with a deck. Atop the deck may be a deckhouse and other superstructures, such as a funnel, derrick, or mast. The line where the hull meets the water surface is called the waterline.
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
In mathematics, the slope or gradient of a line is a number that describes the direction and steepness of the line. Often denoted by the letter m, slope is calculated as the ratio of the vertical change to the horizontal change between two distinct points on the line, giving the same number for any choice of points. A line descending left-to-right has negative rise and negative slope. The line may be physical – as set by a road surveyor, pictorial as in a diagram of a road or roof, or abstract.
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:
In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule. It can also be applied for approximating the length of curves and other approximations.
The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change, liquid vapor. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is:
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
Buoyancy, or upthrust, is a gravitational force, a net upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus, the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object. The pressure difference results in a net upward force on the object. The magnitude of the force is proportional to the pressure difference, and is equivalent to the weight of the fluid that would otherwise occupy the submerged volume of the object, i.e. the displaced fluid.
In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter.
In geometry, a circular segment or disk segment is a region of a disk which is "cut off" from the rest of the disk by a straight line. The complete line is known as a secant, and the section inside the disk as a chord.
Blade element theory (BET) is a mathematical process originally designed by William Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki (1885) to determine the behavior of propellers. It involves breaking a blade down into several small parts then determining the forces on each of these small blade elements. These forces are then integrated along the entire blade and over one rotor revolution in order to obtain the forces and moments produced by the entire propeller or rotor. One of the key difficulties lies in modelling the induced velocity on the rotor disk. Because of this the blade element theory is often combined with momentum theory to provide additional relationships necessary to describe the induced velocity on the rotor disk, producing blade element momentum theory. At the most basic level of approximation a uniform induced velocity on the disk is assumed:
In calculus, the trapezoidal rule is a technique for numerical integration, i.e., approximating the definite integral:
In nuclear physics, the semi-empirical mass formula (SEMF) is used to approximate the mass of an atomic nucleus from its number of protons and neutrons. As the name suggests, it is based partly on theory and partly on empirical measurements. The formula represents the liquid-drop model proposed by George Gamow, which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. It was first formulated in 1935 by German physicist Carl Friedrich von Weizsäcker, and although refinements have been made to the coefficients over the years, the structure of the formula remains the same today.
Babylonian mathematics is the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. With respect to time they fall in two distinct groups: one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. With respect to content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for over a millennium.
In metallurgy, the Scheil-Gulliver equation describes solute redistribution during solidification of an alloy.
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.
In the field of pharmacokinetics, the area under the curve (AUC) is the definite integral of the concentration of a drug in blood plasma as a function of time. In practice, the drug concentration is measured at certain discrete points in time and the trapezoidal rule is used to estimate AUC. In pharmacology, the area under the plot of plasma concentration of a drug versus time after dosage gives insight into the extent of exposure to a drug and its clearance rate from the body.
