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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 enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks ...
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.
In most humid subtropical climates, summer is the wettest season. In summer, the subtropical high pressure cells provide a sultry southernly flow of tropical air with high dew points, and frequent (but brief) convective showers are common.
Some tropical or subtropical regions such as the Philippines or South China are greatly affected by convection cells along a trough. In the mid-latitude westerlies, upper level troughs and ridges often alternate in a high-amplitude pattern. For a trough in the westerlies, the region just west of the trough axis is typically an area of ...
An example of a convergence zone is the Intertropical Convergence Zone (ITCZ), a low pressure area which girdles the Earth at the Equator. [3] Another example is the South Pacific convergence zone that extends from the western Pacific Ocean toward French Polynesia.
Mesoscale convective systems, which can evolve into tropical cyclones, form along areas such as tropical waves or easterly waves which progress westward along monsoon troughs and the Intertropical Convergence Zone in regions of ample low level moisture, convergent surface winds, and divergent winds aloft.
It is straightforward to show that a vector field is path-independent if and only if the integral of the vector field over every closed loop in its domain is zero. Thus the converse can alternatively be stated as follows: If the integral of F over every closed loop in the domain of F is zero, then F is the gradient of some scalar-valued function.