mathlib documentation

analysis.box_integral.divergence_theorem

Divergence integral for Henstock-Kurzweil integral #

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 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

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} [normed_group E] [normed_space E] {n : } [complete_space E] (I : box_integral.box (fin (n + 1))) {i : fin (n + 1)} {f : (fin (n + 1)) → E} {f' : (fin (n + 1)) →L[] E} (hfc : continuous_on f (box_integral.box.Icc I)) {x : fin (n + 1)} (hxI : x box_integral.box.Icc I) {a : E} {ε : } (h0 : 0 < ε) (hε : ∀ (y : fin (n + 1)), y box_integral.box.Icc If y - a - f' (y - x) ε * y - x) {c : ℝ≥0} (hc : I.distortion c) :

Auxiliary lemma for the divergence theorem.

theorem box_integral.has_integral_bot_pderiv {E : Type u} [normed_group E] [normed_space E] {n : } [complete_space E] (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 scontinuous_within_at f (box_integral.box.Icc I) x) (Hd : ∀ (x : fin (n + 1)), x box_integral.box.Icc I \ shas_fderiv_within_at f (f' x) (box_integral.box.Icc I) 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_bot_divergence_of_forall_has_deriv_within_at {E : Type u} [normed_group E] [normed_space E] {n : } [complete_space E] (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 scontinuous_within_at f (box_integral.box.Icc I) x) (Hd : ∀ (x : fin (n + 1)), x box_integral.box.Icc I \ shas_fderiv_within_at f (f' x) (box_integral.box.Icc I) x) :
box_integral.has_integral I (λ (x : fin (n + 1)), ∑ (i : fin (n + 1)), (f' x) (pi.single i 1) i) box_integral.box_additive_map.volume (∑ (i : fin (n + 1)), (box_integral.integral (I.face i) (λ (x : fin n), f (i.insert_nth (I.upper i) x) i) box_integral.box_additive_map.volume - box_integral.integral (I.face i) (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.