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The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials.
In fluid mechanics, Kelvin's circulation theorem states: [1] [2] In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time. The theorem is named after William Thomson, 1st Baron Kelvin who published it in 1869.
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the -plane. We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0.
An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
The number of second derivatives which are independent of each other is relatively small, which means that most material properties can be described in terms of just a few "standard" properties. For the case of a single component system, there are three properties generally considered "standard" from which all others may be derived:
Because the exterior derivative d has the property that d 2 = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The k -th de Rham cohomology (group) is the vector space of closed k -forms modulo the exact k -forms; as noted in the previous section, the Poincaré lemma states that these vector spaces ...
In other words, we can solve for φ(x) everywhere inside a volume where either (1) the value of φ(x) is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of φ(x) is specified on the bounding surface (Neumann boundary conditions). Suppose the problem is to solve for φ(x) inside the ...