Search results
Results from the WOW.Com Content Network
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula. Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative , in terms of which the "geometer's Laplacian" is ...
In physics, the Young–Laplace equation (/ l ə ˈ p l ɑː s /) is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin.
The above-mentioned formula is also known in the engineering community. For instance, a paper written in the Journal of Applied Physics in volume 18, 1947 pages 562-577 shows N.G. De Bruijn and C.J. Boukamp knew of the above relationship. In fact, virtually all the mathematics found in recent papers was already done by Chester Snow.
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...
Paraboloidal coordinates can be useful for solving certain partial differential equations. For instance, the Laplace equation and Helmholtz equation are both separable in paraboloidal coordinates. Hence, the coordinates can be used to solve these equations in geometries with paraboloidal symmetry, i.e. with boundary conditions specified on ...