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This is the divergence theorem. [2] The divergence theorem is employed in any conservation law which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary. [3]
The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: v ⋅ d S = 0 , {\displaystyle \;\;\mathbf {v} \cdot \,d\mathbf {S} =0,}
The following is a proof of half of the theorem for the simplified area D, a type I region where C 1 and C 3 are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the theorem when D is a type II region where C 2 and C 4 are curves connected by horizontal lines (again, possibly of zero ...
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.
Plugging this into the divergence theorem produces Green's theorem, = ^. Suppose that the linear differential operator L is the Laplacian , ∇ 2 , and that there is a Green's function G for the Laplacian.
A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface. Using the first fundamental form, it is possible to define new objects on a regular surface.
Divergence gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate flux by divergence theorem. Curl measures how much "rotation" a vector field has near a point. The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field.