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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.
According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in the same two-dimensional plane are characterized by the existence of a linear relation a κ(t) + b τ(t) = 1 where κ(t) and τ(t) are the curvature and torsion of γ 1 (t) and a and b are ...
In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇ φ is a conservative field . Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the "time derivative" — the rate of change over time — is essential for the precise ...
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function:, where U is an open subset of , that satisfies Laplace's equation, that is, + + + = everywhere on U.
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when