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  2. Integration by parts - Wikipedia

    en.wikipedia.org/wiki/Integration_by_parts

    Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V. [7]

  3. Product rule - Wikipedia

    en.wikipedia.org/wiki/Product_rule

    The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is if it is differentiable.)

  4. Product integral - Wikipedia

    en.wikipedia.org/wiki/Product_integral

    For the case of : [,], the product integral reduces exactly to the case of Lebesgue integration, that is, to classical calculus. Thus, the interesting cases arise for functions f : [ a , b ] → A {\displaystyle f:[a,b]\to A} where A {\displaystyle A} is either some commutative algebra , such as a finite-dimensional matrix field , or if A ...

  5. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    Product rule for multiplication by a scalar ... Integration around a closed curve in the clockwise sense is the negative of the same line integral in the ...

  6. Leibniz integral rule - Wikipedia

    en.wikipedia.org/wiki/Leibniz_integral_rule

    With those tools, the Leibniz integral rule in n dimensions is [4] = () + + ˙, where Ω(t) is a time-varying domain of integration, ω is a p-form, = is the vector field of the velocity, denotes the interior product with , d x ω is the exterior derivative of ω with respect to the space variables only and ˙ is the time derivative of ω.

  7. Wallis' integrals - Wikipedia

    en.wikipedia.org/wiki/Wallis'_integrals

    By means of integration by parts, a reduction formula can be obtained. Using the identity ... we have, by the product rules of equivalents, ...

  8. Integral - Wikipedia

    en.wikipedia.org/wiki/Integral

    Integration, the process of computing an integral, is one of the two fundamental operations of calculus, [a] the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide ...

  9. General Leibniz rule - Wikipedia

    en.wikipedia.org/wiki/General_Leibniz_rule

    The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.