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- Vector Fields: Vector Analysis Developed through its Applications to Engineering and Physics.
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- Vector calculus - Wikipedia.
Home Questions Tags Users Unanswered. Industrial applications of vector calculus [closed] Ask Question. Michael Stachowsky Michael Stachowsky 2. There are 41 matching applications in this category. User Community: Application Center: Vector Calculus. Browse Category: Collision detection between toolholder and workpiece on ball nut grinding Author: Vector space with projections and forces Author: Lenin Araujo Castillo Maple Document.
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Displaying applications There are 41 matching applications in this category. These applications were created using recent versions of Maple. Click here to view our archived Maple-related applications prior to Maple A measure of how "popular" the application is. Includes number of downloads, views, average rating and age. The critical values are the values of the function at the critical points. If the function is smooth , or, at least twice continuously differentiable, a critical point may be either a local maximum , a 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 theorem , all 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 form , and also the cross product , which 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 be taken into account see cross product and handedness for more detail.
Vector calculus can 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 reference , which reflects the fact that vector calculus is invariant under rotations the special orthogonal group SO 3. More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifold , or 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 symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point. Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry , of which vector calculus forms a subset.
Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian yielding harmonic analysis , while curl and cross product do not generalize as directly. From a general point of 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 algebra , uses 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 product , which exists in all dimensions and takes in two vector fields, giving as output a bivector 2-vector field.
Vector Calculus - Application Center
This product yields Clifford algebras as the algebraic structure on vector spaces with an orientation and nondegenerate form. Geometric algebra 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 or k -vector fields, and is widely used in mathematics, particularly in differential geometry , geometric topology , and harmonic analysis , in 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.
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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 k -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 field , which 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 references , related reading or external links , but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations.