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The curl of a 3-dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page. Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector ...
D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do ...
More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given 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 ...
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] 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.
Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.
Any of the operations of vector calculus including gradient, divergence, and curl. Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom . Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus ...