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space with uniformly continuous bounded derivatives, and the case m = ∞
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space of continuous functions with compact support
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id. m times continuously differentiable, and the case m = ∞
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OPERATIONS ON A FUNCTION f
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extension by 0E
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image under permutation of variables
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image under the symmetry x ↦ −x of the variable
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image under separation of variables
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τxf
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translation by x ∈ ℝd
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Rnf
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global regularization
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f ⋄ μ
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function weighted by μ
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f ⋄ ρn
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local regularization
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f * μ
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convolution with μ
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f ⨂ g
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tensor product with g
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f ∘ T
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composition with T
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supp f
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support
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Lf or L ∘ f
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composition with the linear mapping L
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DERIVATIVES OF A FUNCTION f
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fʹ or df/dx
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derivative of a function of a single real variable
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∂if
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partial derivative: ∂if = ∂f/∂xi
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∂β f
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derivative of order
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β
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positive multi-integer: β = (β1,…, βd), βi ≥ 0
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|β|
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differentiability order: |β| = |β11 + … |βd|
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∂0f
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derivative of order 0: ∂0 f = f
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∇f
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gradient: ∇f = (∂1f,…, ∂df)
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df
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differential
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q
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field: q = (q1,…,qd)
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∇· q
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divergence: ∇ · q = ∂1q1 + … ∂dqd
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∇−1q
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primitive that depends continuously on q
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q*
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explicit primitive: q*(x) = fΓ(a, x) q · dℓ
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INTEGRALS AND PATHS
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Cauchy integral
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approximate integral
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surface integral over a sphere
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ƒΓq·dℓ
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line integral of a vector field along a path
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Γ
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path
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[Γ]
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image of a path: [Γ] = {Γ(t) : ti ≤ t ≤ te}
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reverse path
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Γ{a}
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path consisting of a single point
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rectilinear path
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path concatenation
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T
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tube around a path: T = [Γ] + B
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H
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homotopy
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[H]
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image of a homotopy
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