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2) = 1 / 2 n(n − 1) dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have n = 1 / 2 n(n − 1), which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a ...
This is a (dualized) (1 + 1)-dimensional case, for a 1-form (dualized because it is a statement about vector fields). This special case is often just referred to as Stokes' theorem in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as the curl theorem.
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 not exist.
Circulation can be related to curl of a vector field V and, more specifically, to vorticity if the field is a fluid velocity field, =.. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, [4] = = =
As a second-order differential operator, the Laplace operator maps C k functions to C k−2 functions for k ≥ 2.It is a linear operator Δ : C k (R n) → C k−2 (R n), or more generally, an operator Δ : C k (Ω) → C k−2 (Ω) for any open set Ω ⊆ R n.
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...
The Helmholtz decomposition in three dimensions was first described in 1849 [9] by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, [10] [11] which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. [11]
Here the final equality follows by the gradient theorem, since the function f(x) = | x | α+1 is differentiable on R n if α ≥ 1. If α < 1 then this equality will still hold in most cases, but caution must be taken if γ passes through or encloses the origin, because the integrand vector field | x | α − 1 x will fail to be defined there.