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In the study of ordinary differential equations and their associated boundary value problems in mathematics, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville theory.
In geometric measure theory, integration by substitution is used with Lipschitz functions. A bi-Lipschitz function is a Lipschitz function φ : U → R n which is injective and whose inverse function φ −1 : φ(U) → U is also Lipschitz. By Rademacher's theorem, a bi-Lipschitz mapping is differentiable almost everywhere.
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral): ∂ S {\displaystyle {\scriptstyle \partial S}} A ⋅ d ℓ = − {\displaystyle \mathbf {A} \cdot d{\boldsymbol {\ell }}=-} ∂ S {\displaystyle ...
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.
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).
which we can recognize as eigenvalue problems for the operators for and . If T {\displaystyle T} is a compact, self-adjoint operator on the space L 2 [ 0 , l ] {\displaystyle L^{2}[0,l]} along with the relevant boundary conditions, then by the Spectral theorem there exists a basis for L 2 [ 0 , l ] {\displaystyle L^{2}[0,l]} consisting of ...
where ΔU t = U(t) − U(t−). This result can be seen as a precursor to Itô's lemma, and is of use in the general theory of stochastic integration. The final term is ΔU(t)ΔV(t) = d[U, V], which arises from the quadratic covariation of U and V. (The earlier result can then be seen as a result pertaining to the Stratonovich integral.)