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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.
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 ...
Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule; Integral of the secant function; Integral of secant cubed; Arclength ...
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).
When substituted into the above formula, the Leibniz rule as applied for the standard exterior derivative and for the covariant derivative ∇ cancel out the arbitrary choice. [ 6 ] A vector-valued differential 2-form s may be regarded as a certain collection of functions s α ij assigned to an arbitrary local frame of E over a local coordinate ...
The contents of the Differentiation under the integral sign page were merged into Leibniz integral rule on 15 August 2016. For the contribution history and old versions of the redirected page, please see its history ; for the discussion at that location, see its talk page .
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In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point.