Search results
Results from the WOW.Com Content Network
A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line of the integration. Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization ...
Here, θ is the angle between the vectors V and dl. The circulation Γ of a vector field V around a closed curve C is the line integral: [3] [4] =. In a conservative vector field this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken.
In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure.
A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.
An inexact differential is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths ,: [,] such that () = (), () = and In this case, we denote the integrals as | and | respectively to make explicit the path dependence of the change of the quantity we are considering as .
A system moving between two points takes one particular path; other similar paths are not taken. Each path corresponds to a value of the action. An action principle predicts or explains that the particular path taken has a stationary value for the system's action: similar paths near the one taken have very similar action value.
The optical path difference between the paths taken by two identical waves can then be used to find the phase change. Finally, using the phase change, the interference between the two waves can be calculated. Fermat's principle states that the path light takes between two points is the path that has the minimum optical path length.