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Τι (ποιος) είναι quotient manifold - ορισμός

EXPRESSION IN CALCULUS

Newton's quotient; Newton's difference quotient; Difference Quotient; Newton quotient; Fermat's difference quotient

Lie group action

GROUP ACTION OF A LIE GROUP

Quotient manifold

In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.

https://en.wikipedia.org/wiki/Lie_group_action

G2 manifold

SEVEN-DIMENSIONAL RIEMANNIAN MANIFOLD WITH HOLONOMY GROUP CONTAINED IN G2

Joyce manifold; G2-manifold

In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G_2 is one of the five exceptional simple Lie groups.

https://en.wikipedia.org/wiki/G2_manifold

Differentiable manifold

MANIFOLD UPON WHICH IT IS POSSIBLE TO PERFORM CALCULUS (ANY DIFFERENTIABLITY CLASS)

Differential manifold; Smooth manifold; Smooth manifolds; Differentiable manifolds; Manifold/rewrite/differentiable manifold; Differental manifold; Sheaf of smooth functions; Geometric structure; Ambient manifold; Non-smoothable manifold; Curved manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas).

https://en.wikipedia.org/wiki/Differentiable_manifold

Βικιπαίδεια

Difference quotient

In single-variable calculus, the difference quotient is usually the name for the expression

{\frac {f(x+h)-f(x)}{h}}

which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x + h) - x = h in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h).^: 237 The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.

By a slight change in notation (and viewpoint), for an interval [a, b], the difference quotient

{\frac {f(b)-f(a)}{b-a}}

is called the mean (or average) value of the derivative of f over the interval [a, b]. This name is justified by the mean value theorem, which states that for a differentiable function f, its derivative f′ reaches its mean value at some point in the interval. Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)).

Difference quotients are used as approximations in numerical differentiation, but they have also been subject of criticism in this application.

Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h.

The difference quotient is sometimes also called the Newton quotient (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat).

https://en.wikipedia.org/wiki/Difference quotient