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