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In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus.
An example of a solenoidal vector field, (,) = (,) In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: =
In physics, drawings of field lines are mainly useful in cases where the sources and sinks, if any, have a physical meaning, as opposed to e.g. the case of a force field of a radial harmonic. For example, Gauss's law states that an electric field has sources at positive charges , sinks at negative charges, and neither elsewhere, so electric ...
Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series converges to the average of the left and right limits (but see Gibbs phenomenon). The Dirichlet–Dini Criterion states that: [4] if ƒ is 2 π –periodic, locally integrable and satisfies
The divergence of relaxation time at criticality leads to singularities in various collective transport quantities, e.g., the interdiffusivity, shear viscosity, [3] and bulk viscosity . The dynamic critical exponents follow certain scaling relations, viz., z = d + x η {\displaystyle z=d+x_{\eta }} , where d is the space dimension.
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]
A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other ...