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A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. [ 1 ] [ 2 ] It is a plane section of the three-dimensional graph of the function f ( x , y ) {\displaystyle f(x,y)} parallel to the ( x , y ...
The contour integral of a complex function: is a generalization of the integral for real-valued functions. For continuous functions in the complex plane , the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter.
The function to be integrated may be a scalar field or a vector field. 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
As a second example, consider calculating the residues at the singularities of the function = which may be used to calculate certain contour integrals. This function appears to have a singularity at z = 0, but if one factorizes the denominator and thus writes the function as = it is apparent that the singularity at z = 0 is a removable ...
The contour line of an isoquant represents every combination of two inputs which fully maximise a firm's use of resources (such as budget, or time). Full maximisation of resources is usually considered 'efficient'. Efficient allocation of factors of production occur only when two isoquants are tangent to one another.
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proven as a direct consequence of Green's theorem and the fact that the real and imaginary parts of = + must satisfy the Cauchy–Riemann equations in the region bounded by , and moreover in the open neighborhood U of this ...
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [2]