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Irrotational Flow
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected. Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken. Infor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, ''Vector Analysis''. In the conventional form using cross products, vector calculus does not generalize to higher dimensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate System
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function Q in an orthogonal coordinate system. An exact differential is sometimes also called a ''total differential'', or a ''full differential'', or, in the study of differential geometry, it is termed an exact form. The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics. Overview Definition Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type :A(x,y,z) \,dx + B(x,y,z) \,dy + C(x,y,z) \,dz is called a differential form. This form is called ''exact'' on an open domain D \subset \mathbb^3 in spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area. The first part of the theorem, the first fundamental theorem of calculus, states that for a function , an antiderivative or indefinite integral may be obtained as the integral of over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, :h'(x) = f'(g(x)) g'(x). or, equivalently, :h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as :\frac = \frac \cdot \frac, and : \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing the instantaneous rate of cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as W=\mathbf\cdot\mathbf, have natural continuous analogues in terms of line integrals, in this case \textstyle W = \int_L \mathbf(\mathbf)\cdot d\mathbf, which computes the work d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. For as a differentiable function and as any continuous curve in which starts at a point and ends at a point , then \int_ \nabla\varphi(\mathbf)\cdot \mathrm\mathbf = \varphi\left(\mathbf\right) - \varphi\left(\mathbf\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a ''conservative'' force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solenoidal Field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a Helmholtz decomposition#Longitudinal and transverse fields, transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf = 0. A common way of expressing this property is to say that the field has no sources or sinks.This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017. Properties The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational ( curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz. As an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the form -\nabla \phi + \nabla \times \mathbf, where \phi is a scalar field called "scalar potential", and is a vector field, called a vector potential. Statement of the theorem Let \mathbf be a vector field on a bounded domain V\subseteq\mathbb^3, which is twice continuousl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity. A ''scalar potential'' is a fundamental concept in vector analysis and physics (the adjective ''scalar'' is frequently omitted if there is no danger of confusion with ''vector potential''). The scalar potential is an example of a scalar field. Given a vector field , the scalar potential is defined such that: : \mathbf = -\nabla P = - \left( \frac, \frac, \frac \right), where is the gradient of and the second part of the equation is minus the gradient for a function of the Cartesian coordinates . In some cases, mathematicians may use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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