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The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on ...
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space ...
Stokes' theorem, [1] also known as the Kelvin–Stokes theorem [2] [3] after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, [4] is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the ...
Line integral convolution. In scientific visualization, line integral convolution ( LIC) is a method to visualize a vector field (such as fluid motion) at high spatial resolutions. [1] The LIC technique was first proposed by Brian Cabral and Leith Casey Leedom in 1993.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . [1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed ...
In physics, circulation is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field. Circulation was first used independently by Frederick Lanchester, Martin Kutta and Nikolay Zhukovsky. [citation needed]
The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with n = 2 {\displaystyle n=2} ) once we identify a vector field with a 1-form using the metric on ...
Calculus. In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in ). In one dimension, it is equivalent to the fundamental theorem of calculus.