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The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called C r-curves and are central objects studied in the differential geometry of curves.
A smooth function f : M → ℝ on a real differentiable manifold M is a 0-form. The exterior derivative of this 0 -form is the 1 -form df . When an inner product ·,· is defined, the gradient ∇ f of a function f is defined as the unique vector in V such that its inner product with any element of V is the directional derivative of f along ...
The most basic non-trivial differential one-form is the "change in angle" form . This is defined as the derivative of the angle "function" (,) (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function.
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average". [51] Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is differential algebra.
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
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 : ′ = ′ = ′ (()).
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