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#Divergence in curved space pdf
#Divergence in curved space series

The volume rate of flow of liquid inward through the surface S equals the rate of liquid removed by the sink. Similarly if there is a sink or drain inside S, such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain. The flux outward through S equals the volume rate of flow of fluid into S from the pipe. This will cause a net outward flow through the surface S. However if a source of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. If the liquid is moving, it may flow into the volume at some points on the surface S and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the net flux of liquid out of the volume is zero. Since liquids are incompressible, the amount of liquid inside a closed volume is constant if there are no sources or sinks inside the volume then the flux of liquid out of S is zero. The flux of liquid out of the volume is equal to the volume rate of fluid crossing this surface, i.e., the surface integral of the velocity over the surface. Consider an imaginary closed surface S inside a body of liquid, enclosing a volume of liquid. A moving liquid has a velocity-a speed and a direction-at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Vector fields are often illustrated using the example of the velocity field of a fluid, such as a gas or liquid.

7.1 Differential and integral forms of physical laws.

In two dimensions, it is equivalent to Green's theorem. In one dimension, it is equivalent to integration by parts. However, it generalizes to any number of dimensions. In these fields, it is usually applied in three dimensions. The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. Intuitively, it states that the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. (Riemann, Ricci tensors and scalar curvature) and linear in their secondĬovariant derivatives.Theorem in calculus which relates the flux of closed surfaces to divergence over their volume Its divergent part canīe isolated, and a concise formula is here obtained: by dimensional analysisĪnd combinatorics, there are two kinds of terms: quadratic in curvature tensors

#Divergence in curved space series

Obtains a series expansion for the stress-energy tensor. Among these, the stress-energy tensor isĮxpressed in terms of second covariant derivatives of the Hadamard Greenįunction, which is also closely linked to the effective action therefore one The point-splitting method is then applied, since it is a valuable tool for Space-time is studied by using the Fock-Schwinger-DeWitt asymptotic expansion

#Divergence in curved space pdf

Authors: Roberto Niardi, Giampiero Esposito, Francesco Tramontano Download PDF Abstract: In this paper the Feynman Green function for Maxwell's theory in curved