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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-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.
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.
The gradient of a function is obtained by raising the index of the differential , whose components are given by: ∇ i ϕ = ϕ ; i = g i k ϕ ; k = g i k ϕ , k = g i k ∂ k ϕ = g i k ∂ ϕ ∂ x k {\displaystyle \nabla ^{i}\phi =\phi ^{;i}=g^{ik}\phi _{;k}=g^{ik}\phi _{,k}=g^{ik}\partial _{k}\phi =g^{ik}{\frac {\partial \phi }{\partial x^{k}}}}
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". [52] Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is differential algebra.
The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R n. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold
The q-derivative of a function is defined by the formula () = () (). For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative, thus the q-derivative may be viewed as its q-deformation.
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.
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