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In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [ l ] is defined as the linear part of the change in the functional, and the second variation [ m ] is defined as the quadratic part.
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. [1] It is one of the two traditional divisions of calculus, the other being integral calculus —the study of the area beneath a curve.
A time scale (or measure chain) is a closed subset of the real line. The common notation for a general time scale is T {\displaystyle \mathbb {T} } . The two most commonly encountered examples of time scales are the real numbers R {\displaystyle \mathbb {R} } and the discrete time scale h Z {\displaystyle h\mathbb {Z} } .
In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus.The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus.
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 difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was first created by Charles Babbage . The name difference engine is derived from the method of finite differences , a way to interpolate or tabulate functions by using a small set of polynomial co-efficients.
In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold. [10] Indeed, let f : U → R r {\displaystyle f:U\to \mathbb {R} ^{r}} be such a smooth map from an open subset of R n {\displaystyle \mathbb {R} ^{n}} (since the result is local, there is no loss of ...