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In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
Sum rule in integration; Constant factor rule in integration; Linearity of integration; Arbitrary constant of integration; Cavalieri's quadrature formula; Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign ...
Reynolds transport theorem can be expressed as follows: [1] [2] [3] = + () in which n(x,t) is the outward-pointing unit normal vector, x is a point in the region and is the variable of integration, dV and dA are volume and surface elements at x, and v b (x,t) is the velocity of the area element (not the flow velocity).
For example, suppose we want to find the integral ∫ 0 ∞ x 2 e − 3 x d x . {\displaystyle \int _{0}^{\infty }x^{2}e^{-3x}\,dx.} Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it.
In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a formula for a differentiable function F(x) such that
The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a separable Hilbert space ( H , , ) equipped with a Gaussian measure γ . As stated in the article on the Vitali covering theorem , the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces.
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let f {\displaystyle f} and g {\displaystyle g} be n {\displaystyle n} -times differentiable functions. The base case when n = 1 {\displaystyle n=1} claims that: ( f g ) ′ = f ′ g + f g ′ , {\displaystyle (fg)'=f'g+fg',} which is the usual product rule and is known ...
The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A ⋅( B × C ) = ( A × B )⋅ C we may derive A ⋅(∇× C ) = ( A ×∇)⋅ C but not ∇⋅( B × C ) = (∇× B ...
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