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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 ...
Calculus. 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 ...
The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation curl F is more common in North America.
A form of the mean value theorem, where a < ξ < b, can be applied to the first and last integrals of the formula for Δ φ above, resulting in. Dividing by Δ α, letting Δ α → 0, noticing ξ1 → a and ξ2 → b and using the above derivation for yields. This is the general form of the Leibniz integral rule.
Kronecker delta. In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets : For example, because , whereas because . The Kronecker delta appears naturally in many areas of ...
Calculus. In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields ( tensors that may vary over a manifold, e.g. in spacetime ). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, [ 1] it was used by Albert Einstein to develop his general theory of relativity.
Image of the Large Magellanic Cloud, one of the nearest galaxies to our Milky Way, created with LIC. 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 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]