divergence theorem वाक्य
उदाहरण वाक्य
मोबाइल
- The divergence theorem is valid for piecewise smooth domains " D ", and hence " D " needs to be piecewise smooth.
- This theorem is a special case of the divergence theorem, and is essentially the higher dimensional equivalent of integration by parts with and the gradient of replacing and.
- This classical statement, along with the classical divergence theorem, the fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above.
- Only the " differential form " of the equations is given, not the " integral form "; to get the integral forms apply the divergence theorem or Kelvin Stokes theorem.
- The third equality follows by the divergence theorem and shows, again, a sum ( or, in this case, an integral ) of outward normal derivatives over all boundary locations.
- More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
- The divergence theorem states that any such continuity equation can be written in a differential form ( in terms of a divergence ) and an integral form ( in terms of a flux ).
- The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:
- The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:
- The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem.
- Since ? is a dummy variable of integration, and since the change in the boundary ? is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form
- The Gauss s divergence theorem shows that the diffusion equation is valid in the solid, liquid and gas states in every material as a material conservation law, if there is no sink and source in the given diffusion system.
- The divergence theorem is employed in any conservation law which states that the volume total of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.
- Then, as first demonstrated by Newton, and can easily be demonstrated using the divergence theorem, the acceleration of gravity at any given distance R from the center of the sphere depends only upon the total mass contained within R.
- The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes'theorem simultaneously generalizes the three theorems of vector calculus : the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem.
- If u and \ vec f are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and \ partial / \ partial t to get a conservation law for the quantity u in the general form
- The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region ( which is expressed by the divergence ).
- The fundamental theorems of integral calculus in several variables namely Green's theorem, the divergence theorem, and Stokes'theorem & mdash; generalize to a theorem ( also called Stokes'theorem ) relating the exterior derivative and integration over submanifolds.
- As a result of the divergence theorem, a host of physical laws can be written in both a differential form ( where one quantity is the divergence of another ) and an integral form ( where the flux of one quantity through a closed surface is equal to another quantity ).
- In particular, the key fact is that, for vector fields \ mathbf { F } and \ mathbf { G }, integration by parts ( or the divergence theorem ) over a volume " V " enclosed by a surface " S " gives the identity:
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