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Surface Area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri Lebesgue
Henri Lebesgue
and Hermann Minkowski
Hermann Minkowski
at the turn of the twentieth century
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Elephant
Elephants are large mammals of the family Elephantidae
Elephantidae
and the order Proboscidea. Three species are currently recognised: the African bush elephant ( Loxodonta
Loxodonta
africana), the African forest elephant
African forest elephant
(L. cyclotis), and the Asian elephant
Asian elephant
( Elephas
Elephas
maximus). Elephants are scattered throughout sub-Saharan Africa, South Asia, and Southeast Asia
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Square Pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.Contents1 Johnson solid
Johnson solid
(J1) 2 Other square pyramids 3 Related polyhedra and honeycombs3.1 Dual polyhedron4 Topology 5 Examples 6 References 7 External links Johnson solid
Johnson solid
(J1)[edit] If the sides are all equilateral triangles, the pyramid is one of the Johnson solids (J1). The 92 Johnson solids were named and described by Norman Johnson in 1966. A Johnson solid
Johnson solid
is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1] The Johnson square pyramid can be characterized by a single edge-length parameter a
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Dihedral Angle
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge
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Reaction Rate
The reaction rate or rate of reaction is the speed at which reactants are converted into products. For example, the oxidative rusting of iron under Earth's atmosphere
Earth's atmosphere
is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. Chemical kinetics
Chemical kinetics
is the part of physical chemistry that studies reaction rates
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Iron
Iron
Iron
is a chemical element with symbol Fe (from Latin: ferrum) and atomic number 26. It is a metal in the first transition series. It is by mass the most common element on Earth, forming much of Earth's outer and inner core. It is the fourth most common element in the Earth's crust. Its abundance in rocky planets like Earth
Earth
is due to its abundant production by fusion in high-mass stars, where it is the last element to be produced with release of energy before the violent collapse of a supernova, which scatters the iron into space. Like the other group 8 elements, ruthenium and osmium, iron exists in a wide range of oxidation states, −2 to +7, although +2 and +3 are the most common. Elemental iron occurs in meteoroids and other low oxygen environments, but is reactive to oxygen and water. Fresh iron surfaces appear lustrous silvery-gray, but oxidize in normal air to give hydrated iron oxides, commonly known as rust
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Triangular Prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms
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Combustion
Combustion
Combustion
/kəmˈbʌs.tʃən/, or burning,[1] is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion
Combustion
in a fire produces a flame, and the heat produced can make combustion self-sustaining. Combustion
Combustion
is often a complicated sequence of elementary radical reactions. Solid fuels, such as wood and coal, first undergo endothermic pyrolysis to produce gaseous fuels whose combustion then supplies the heat required to produce more of them. Combustion
Combustion
is often hot enough that light in the form of either glowing or a flame is produced. A simple example can be seen in the combustion of hydrogen and oxygen into water vapor, a reaction commonly used to fuel rocket engines
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Mitochondrion
1 Outer membrane1.1 Porin2 Intermembrane space2.1 Intracristal space 2.2 Peripheral space3 Lamella3.1 Inner membrane3.11 Inner boundary membrane 3.12 Cristal membrane3.2 Matrix 3.3 Cristæ4 Mitochondrial DNA 5 Matrix granule 6 Ribosome 7 ATP synthaseThe mitochondrion (plural mitochondria) is a double-membrane-bound organelle found in most eukaryotic organisms. Some cells in some multicellular organisms may however lack them (for example, mature mammalian red blood cells). A number of unicellular organisms, such as microsporidia, parabasalids, and diplomonads, have also reduced or transformed their mitochondria into other structures.[1] To date, only one eukaryote, Monocercomonoides, is known to have completely lost its mitochondria.[2] The word mitochondrion comes from the Greek μίτος, mitos, "thread", and χονδρίον, chondrion, "granule"[3] or "grain-like"
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Fractal
Mandelbrot set: Self-similarity
Self-similarity
illustrated by image enlargements. This panel, no magnification.The same fractal as above, magnified 6-fold. Same patterns reappear, making the exact scale being examined difficult to determine.The same fractal as above, magnified 100-fold.The same fractal as above, magnified 2000-fold, where the Mandelbrot set fine detail resembles the detail at low magnification.In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects. Artificially created fractals commonly exhibit similar patterns at increasingly small scales.[1] It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger sponge.[2] Fractals can also be nearly the same at different levels
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Hermann Schwarz
Karl Hermann Amandus Schwarz (German: [ʃvaʁts]; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.Contents1 Life 2 Work 3 Publications 4 Notes 5 External linksLife[edit] Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). He was married to Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer[1] and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of Moses Mendelssohn). Schwarz and Kummer had six children.[1] Schwarz originally studied chemistry in Berlin
Berlin
but Ernst Eduard Kummer and Karl Theodor Wihelm Weierstrass persuaded him to change to mathematics.[2] He received his Ph.D
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Cellular Respiration
Cellular respiration
Cellular respiration
is a set of metabolic reactions and processes that take place in the cells of organisms to convert biochemical energy from nutrients into adenosine triphosphate (ATP), and then release waste products.[1] The reactions involved in respiration are catabolic reactions, which break large molecules into smaller ones, releasing energy in the process, as weak so-called "high-energy" bonds are replaced by stronger bonds in the products. Respiration is one of the key ways a cell releases chemical energy to fuel cellular activity. Cellular respiration
Cellular respiration
is considered an exothermic redox reaction which releases heat
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Continuously Differentiable
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps. More generally, if x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0, f(x0))
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Euclidean Group
In mathematics, the Euclidean group
Euclidean group
E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements are the isometries associated with the Euclidean distance, and are called Euclidean isometries, Euclidean transformations or Rigid transformations. Euclidean isometries are classified into direct isometries and indirect isometries, an indirect isometry being an isometry that transforms any object into its mirror image
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Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.[1] More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object
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Surface (topology)
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space
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