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analysis.box_integral.divergence_theorem

# Divergence integral for Henstock-Kurzweil integral #

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In this file we prove the Divergence Theorem for a Henstock-Kurzweil style integral. The theorem says the following. Let f : ℝⁿ → Eⁿ be a function differentiable on a closed rectangular box I with derivative f' x : ℝⁿ →L[ℝ] Eⁿ at x ∈ I. Then the divergence λ x, ∑ k, f' x eₖ k, where eₖ = pi.single k 1 is the k-th basis vector, is integrable on I, and its integral is equal to the sum of integrals of f over the faces of I taken with appropriate signs.

To make the proof work, we had to ban tagged partitions with “long and thin” boxes. More precisely, we use the following generalization of one-dimensional Henstock-Kurzweil integral to functions defined on a box in ℝⁿ (it corresponds to the value box_integral.integration_params.GP = ⊥ of box_integral.integration_params in the definition of box_integral.has_integral).

We say that f : ℝⁿ → E has integral y : E over a box I ⊆ ℝⁿ if for an arbitrarily small positive ε and an arbitrarily large c, there exists a function r : ℝⁿ → (0, ∞) such that for any tagged partition π of I such that

• π is a Henstock partition, i.e., each tag belongs to its box;
• π is subordinate to r;
• for every box of π, the maximum of the ratios of its sides is less than or equal to c,

the integral sum of f over π is ε-close to y. In case of dimension one, the last condition trivially holds for any c ≥ 1, so this definition is equivalent to the standard definition of Henstock-Kurzweil integral.

## Tags #

Henstock-Kurzweil integral, integral, Stokes theorem, divergence theorem

theorem box_integral.norm_volume_sub_integral_face_upper_sub_lower_smul_le {E : Type u} [ E] {n : } (I : box_integral.box (fin (n + 1))) {i : fin (n + 1)} {f : (fin (n + 1) ) E} {f' : (fin (n + 1) ) →L[] E} (hfc : ) {x : fin (n + 1) } (hxI : x ) {a : E} {ε : } (h0 : 0 < ε) (hε : (y : fin (n + 1) ), f y - a - f' (y - x) ε * y - x) {c : nnreal} (hc : I.distortion c) :

Auxiliary lemma for the divergence theorem.

theorem box_integral.has_integral_GP_pderiv {E : Type u} [ E] {n : } (I : box_integral.box (fin (n + 1))) (f : (fin (n + 1) ) E) (f' : (fin (n + 1) ) ((fin (n + 1) ) →L[] E)) (s : set (fin (n + 1) )) (hs : s.countable) (Hs : (x : fin (n + 1) ), x s ) (Hd : (x : fin (n + 1) ), x (f' x) x) (i : fin (n + 1)) :

If f : ℝⁿ⁺¹ → E is differentiable on a closed rectangular box I with derivative f', then the partial derivative λ x, f' x (pi.single i 1) is Henstock-Kurzweil integrable with integral equal to the difference of integrals of f over the faces x i = I.upper i and x i = I.lower i.

More precisely, we use a non-standard generalization of the Henstock-Kurzweil integral and we allow f to be non-differentiable (but still continuous) at a countable set of points.

TODO: If n > 0, then the condition at x ∈ s can be replaced by a much weaker estimate but this requires either better integrability theorems, or usage of a filter depending on the countable set s (we need to ensure that none of the faces of a partition contain a point from s).

theorem box_integral.has_integral_GP_divergence_of_forall_has_deriv_within_at {E : Type u} [ E] {n : } (I : box_integral.box (fin (n + 1))) (f : (fin (n + 1) ) fin (n + 1) E) (f' : (fin (n + 1) ) ((fin (n + 1) ) →L[] fin (n + 1) E)) (s : set (fin (n + 1) )) (hs : s.countable) (Hs : (x : fin (n + 1) ), x s ) (Hd : (x : fin (n + 1) ), x (f' x) x) :
(λ (x : fin (n + 1) ), finset.univ.sum (λ (i : fin (n + 1)), (f' x) 1) i)) box_integral.box_additive_map.volume (finset.univ.sum (λ (i : fin (n + 1)), (λ (x : fin n ), f (i.insert_nth (I.upper i) x) i) box_integral.box_additive_map.volume - (λ (x : fin n ), f (i.insert_nth (I.lower i) x) i) box_integral.box_additive_map.volume))

Divergence theorem for a Henstock-Kurzweil style integral.

If f : ℝⁿ⁺¹ → Eⁿ⁺¹ is differentiable on a closed rectangular box I with derivative f', then the divergence ∑ i, f' x (pi.single i 1) i is Henstock-Kurzweil integrable with integral equal to the sum of integrals of f over the faces of I taken with appropriate signs.

More precisely, we use a non-standard generalization of the Henstock-Kurzweil integral and we allow f to be non-differentiable (but still continuous) at a countable set of points.