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Tensor Derivative (continuum Mechanics)
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. The directional derivative provides a systematic way of finding these derivatives. Derivatives with respect to vectors and second-order tensors The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken. Derivatives of scalar valued functions of vectors Let ''f''(v) be a real valued function of the vector v. Then the derivative of ''f''(v) with respect to v (or at v) is the vector defined through its dot product with any vector u being \frac\cdot\mathbf = Df(\mathbf) mathbf= \left frac~f(\mathbf + \alpha~\mathbf)\right for all vectors u. The above dot product yields a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Directional Derivative
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction specified by v. The directional derivative of a scalar function ''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: \begin \nabla_(\mathbf) &=f'_\mathbf(\mathbf)\\ &=D_\mathbff(\mathbf)\\ &=Df(\mathbf)(\mathbf)\\ &=\partial_\mathbff(\mathbf)\\ &=\mathbf\cdot\\ &=\mathbf\cdot \frac.\\ \end It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Curvilinear Coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name ''curvilinear coordinates'', coined by the French mathematician Gabriel Lamé, Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are Cylindrical coordinate system, cylindrical and spherical coordinates, spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example ''z'' = 0 defines the ''x''-''y'' plane. In the same space, the coordinate surface ''r'' = 1 in spherical coordinates i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Covariant Derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between two random variables or data sets * Autocovariance, the covariance of a signal with a time-shifted version of itself * Covariance function, a function giving the covariance of a random field with itself at two locations Algebra and geometry * A covariant (invariant theory) is a bihomogeneous polynomial in and the coefficients of some homogeneous form in that is invariant under some group of linear transformations. * Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis ** Covariant transformation, a rule that describes how certain physical entities change under a change of coordinate system * Covariance and contravariance of functors, properties of functors * General covariance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Divergence Theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region". The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Permutation Symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers , for some positive integer . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon or , or less commonly the Latin lower case . Index notation allows one to display permutations in a way compatible with tensor analysis: \varepsilon_ where ''each'' index takes values . There are indexed values of , which can be arranged into an -dimensional array. The key defining property of the symbol is ''total antisymmetry'' in the indices. When any two indices are interchanged, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Curl (mathematics)
In vector calculus, the curl, also known as rotor, is a vector operator that describes the Differential (infinitesimal), infinitesimal Circulation (physics), circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector (geometry), vector whose length and direction denote the Magnitude (mathematics), magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called irrotational. The curl is a form of derivative, differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Kelvin–Stokes theorem, Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation is more common in North America. In the rest of the world, particularly in 20th century scientific li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cylindrical Coordinates
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in a solid ball versus sphere surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term "cylinder" could refer to either of these or to an even more specialized object, the '' right circular cylinder''. Types The definitions and results in this section are taken from the 1913 text ''Plane and Solid Geometry'' by George A. Wentworth and David Eugene Smith . A ' is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ricci Calculus
Ricci () is an Italian surname. Notable Riccis Arts and entertainment * Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin * Christina Ricci (born 1980), American actress * Clara Ross Ricci (1858-1954), British composer * Federico Ricci (1809–77), Italian composer * Franco Maria Ricci (1937–2020), Italian art publisher * Italia Ricci (born 1986), Canadian actress * Jason Ricci (born 1974), American blues harmonica player * Lella Ricci (1850–71), Italian singer * Luigi Ricci (1805–59), Italian composer * Luigi Ricci (1893–1981), Italian vocal coach * Luigi Ricci-Stolz (1852–1906), Italian composer * Marco Ricci (1676–1730), Italian Baroque painter * Nahéma Ricci, Canadian actress * Nina Ricci (designer) (1883–1970), French fashion designer * Nino Ricci (born 1959), Canadian novelist * Regolo Ricci (born 1955), Canadian painter and illustrator * Romina Ricci (born 1978), Argentine actress * Ruggiero Ricci (1918–2012 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to area.) More precisely, the divergence at a point is the rate that the flow of the vector field modifies a volume about the point ''in the limit'', as a small volume shrinks down to the point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value. Physical interpretation of divergence In physical terms, the divergence of a vector field is the extent to which the vector fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cylindrical Coordinate
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions around a main axis (a chosen directed line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpendicular distance from the main axis; the point signed distance ''z'' along the main axis from a chosen origin; and the plane angle of the point projection on a reference plane (passing through the origin and perpendicular to the main axis) The main axis is variously called the ''cylindrical'' or ''longitudinal'' axis. The auxiliary axis is called the ''polar axis'', which lies in the reference plane, starting at the origin, and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are called ''radial lines''. The distance from the axis may be called the ''radial distance'' or ''radius'', while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''. The r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Christoffel Symbol
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each " frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group . As a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |