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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.
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. [1] Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea. [2] [3] [4]
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
The exterior derivative is an operation on differential forms that, given a k-form , produces a (k+1)-form . This operation extends the differential of a function (a function can be considered as a 0 -form, and its differential is d f ( x ) = f ′ ( x ) d x {\displaystyle df(x)=f'(x)dx} ).
for the first derivative, for the second derivative, for the third derivative, and for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken.
The metric tensor (,) induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat ♭ and sharp ♯.A section () corresponds to the unique one-form ♭ such that for all sections (), we have:
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.