Gradient, Divergence, and Curl. The gradient, divergence, and curl are the result of applying the Del operator to various kinds of functions: The Gradient is what. For any function q in H1(Ω◦), grad q is the gradient of q in the sense of .. domaines des opérateurs divergence et rotationnel avec trace nulle. – Buy Analyse Vectorielle: Thorme De Green, Gradient, Divergence, Oprateur Laplacien, Rotationnel, Champ De Vecteurs, Nabla book online at best .
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Vector calculusor vector analysisis a branch of mathematics concerned with differentiation and integration of vector fieldsprimarily in 3-dimensional Euclidean space R 3.
Vector calculus plays an important role in differential geometry and in the hradient of partial differential equations. It is used extensively in physics and engineeringespecially in the description of electromagnetic fieldsgravitational 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 book, Vector Analysis. In the conventional form using cross productsvector calculus does not generalize to higher dimensions, while the alternative rotationnel of geometric algebrawhich uses exterior products does generalize, as discussed totationnel.
A scalar field associates a scalar value to every point in a space.
The scalar may either be a mathematical number or a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such gradiwnt the Higgs field. These fields are the subject of scalar field theory.
A vector field is an assignment of a vector to each point in a subset of space. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some forcesuch as the magnetic or gravitational force, as it changes from point to point.
In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: This distinction is clarified and elaborated in geometric algebra, as described below. The algebraic non-differential operations in vector calculus are referred to as vector algebrabeing defined for a vector space and then globally applied to a vector field.
The basic algebraic operations consist of:. Also commonly used are the two triple products:. The three basic vector operators are:. A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration. The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions:. Linear approximations are used to replace complicated functions with linear functions that are almost the same.
For a continuously differentiable function of several real variablesa point P that is a set of values for the input variables, which is viewed as a point in R n is critical if all of the partial derivatives of the function are zero at Por, equivalently, if its gradient is zero.
The critical values are the values of the function at the critical points. If the function is smoothor, at least twice continuously differentiable, a critical point may be either a local maximuma local minimum or a saddle point. The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.
By Fermat’s theoremall local maxima and minima of a differentiable function occur at critical points.
Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros. These structures give rise to a volume formand also the cross productwhich is used pervasively in vector calculus. The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the coordinate system to rotaationnel taken into account see cross product and handedness for more detail.
Vector calculus rotationnell be defined on other 3-dimensional real vector spaces if they have an inner product or more generally a symmetric nondegenerate form and an orientation; note that this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates a frame of referencewhich reflects the fact that vector calculus is invariant under rotations the rotahionnel orthogonal group SO 3.
More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifoldor more generally pseudo-Riemannian manifold. This structure simply means that the tangent space at each point has an inner product more generally, a symmetric nondegenerate form and an orientation, or more globally that there is a rotatiknnel nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.
Most of divergenxe analytic results are easily understood, in a more general form, using the machinery of differential geometryof which vector calculus forms a subset. Grad and div generalize royationnel to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian yielding harmonic analysiswhile curl and cross product do not generalize as directly.
From a general point gravient view, the various fields in 3-dimensional vector calculus are uniformly seen as being k -vector fields: The generalization of grad and div, and how curl may be generalized is elaborated at Curl: There are two important alternative generalizations of vector calculus.
The first, geometric algebrauses k -vector fields instead of vector fields in 3 or fewer dimensions, every k -vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions. This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the exterior productwhich exists in all dimensions and takes in two vector fields, giving as output a bivector 2-vector field.
This product yields Clifford algebras as the algebraic structure on vector spaces with an orientation and nondegenerate form. Geometric rottionnel is mostly used in generalizations of physics and other applied fields to higher dimensions.
The second generalization uses differential forms k -covector fields instead of vector fields roattionnel k -vector fields, and is widely used in mathematics, gradidnt in differential geometrygeometric topologyand harmonic analysisin particular yielding Hodge theory on oriented pseudo-Riemannian manifolds.
From this point of view, grad, curl, and div correspond to the exterior derivative of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes’ theorem.
From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear. From the point of view of geometric algebra, vector calculus implicitly identifies divergemce -vector fields with vector fields or scalar functions: From the point of view of differential forms, vector calculus implicitly identifies k -forms with scalar fields or vector fields: Thus for example the curl naturally takes as input a vector field or 1-form, but naturally has as output a 2-vector field or 2-form hence pseudovector fieldwhich is then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in the curl of a vector field in higher dimensions not having as output a vector field.
From Wikipedia, the free encyclopedia. Not to be confused with Geometric calculus or Matrix calculus. This article includes a list of referencesrelated reading or external linksbut its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations.
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Mean value theorem Rolle’s theorem. Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor’s theorem. Fractional Malliavin Stochastic Variations. GradientDivergenceCurl mathematicsand Laplacian.
Real-valued function Function of a real variable Real multivariable function Vector calculus identities Del in cylindrical and spherical coordinates Directional derivative Irrotational vector field Solenoidal vector field Laplacian vector field Helmholtz decomposition Orthogonal coordinates Skew coordinates Curvilinear coordinates Tensor. Vector Analysis Versus Vector Calculus. Uses authors parameter link.
Helmholtz decomposition – Wikipedia
Integral Lists of integrals. Specialized Fractional Malliavin Stochastic Variations. Glossary of calculus Glossary of calculus. Measures the rottionnel between the value of the scalar field with its average on infinitesimal balls.
Measures the difference between the value of the vector field with its average on infinitesimal balls. The line integral of the gradient of a scalar field over a curve L is equal rotationnnel the change in the scalar field between the endpoints p and q of the curve.