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For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture). The arcsine is a partial inverse of the sine function.
In mathematics, inverse mapping theorem may refer to: the inverse function theorem on the existence of local inverses for functions with non-singular derivatives the bounded inverse theorem on the boundedness of the inverse for invertible bounded linear operators on Banach spaces
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
Similarly, if is a smooth function on an open set in , then the same formula defines a smooth function on the open set (). (In the language of sheaves , pullback defines a morphism from the sheaf of smooth functions on N {\displaystyle N} to the direct image by ϕ {\displaystyle \phi } of the sheaf of smooth functions on M {\displaystyle M} .)
Another example is the application of conformal mapping technique for solving the boundary value problem of liquid sloshing in tanks. [ 19 ] If a function is harmonic (that is, it satisfies Laplace's equation ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} ) over a plane domain (which is two-dimensional), and is transformed via a conformal map to ...
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map,