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The word flux comes from Latin: fluxus means "flow", and fluere is "to flow". [2] As fluxion, this term was introduced into differential calculus by Isaac Newton.. The concept of heat flux was a key contribution of Joseph Fourier, in the analysis of heat transfer phenomena. [3]
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.
Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of sources or sinks of that quantity. 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). [12]
It can be used to calculate directional derivatives of scalar functions or normal directions. 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.
[note 1] If the flow is assumed to be incompressible then is solenoidal, that is, the divergence is zero: =, and the above equation reduces to + = In particular, if the flow is steady , then [ 2 ] u ⋅ ∇ ψ = 0 {\displaystyle {\mathbf {u} }\cdot \nabla \psi =0} which shows that ψ {\displaystyle \psi } is constant along a streamline .
Given an open subset U of R n and a subinterval I of R, one says that a function u : U × I → R is a solution of the heat equation if = + +, where (x 1, ..., x n, t) denotes a general point of the domain.
We assume that is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension , we can apply the divergence theorem, i.e. =, and substitute for the volume integral of the divergence with the values of () evaluated at the cell surface (edges / and + /) of the finite volume as follows:
In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus: [1]: 543ff Gradient theorem; Stokes' theorem; Divergence theorem; Green's theorem.