Continuous Functions. Jacques Simon

space with uniformly continuous bounded derivatives, and the case m = ∞ space of continuous functions with compact support id. m times continuously differentiable, and the case m = ∞

      OPERATIONS ON A FUNCTION f

	extension by 0E
	image under permutation of variables
	image under the symmetry x ↦ −x of the variable
	image under separation of variables
τxf	translation by x ∈ ℝ^d
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

	Cauchy integral
	approximate integral
	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}
	reverse path
Γ{a}	path consisting of a single point
	rectilinear path
	path concatenation
T	tube around a path: T = [Γ] + B
H	homotopy
[H]	image of a homotopy