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In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
The further fact that it is a differential 3-form valued in E asserts the full anti-symmetry in i, j, k and is directly verified from the above formula and the contextual assumption that s is a vector-valued differential 2-form, so that s α ij = −s α ji.
It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation. If u = u(x) and du = u ′ (x) dx, while v = v(x) and dv = v ′ (x) dx, then integration by parts states that:
The covariant derivative is a generalization of the directional derivative from vector calculus.As with the directional derivative, the covariant derivative is a rule, , which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. [7]
In particular, Γ does not involve any derivatives on u or v. In this approach, Γ must transform in a prescribed manner when the coordinate system φ is changed to a different coordinate system. This transformation is not tensorial , since it involves not only the first derivative of the coordinate transition, but also its second derivative .
In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus, differential geometry, and differential forms. [1]
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
Combining derivatives of different variables results in a notion of a partial differential operator. The linear operator which assigns to each function its derivative is an example of a differential operator on a function space. By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus.
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