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linear_algebra.affine_space.slope

Slope of a function #

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In this file we define the slope of a function f : k → PE taking values in an affine space over k and prove some basic theorems about slope. The slope function naturally appears in the Mean Value Theorem, and in the proof of the fact that a function with nonnegative second derivative on an interval is convex on this interval.

Tags #

affine space, slope

def slope {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b : k) :

E

slope f a b = (b - a)⁻¹ • (f b -ᵥ f a) is the slope of a function f on the interval [a, b]. Note that slope f a a = 0, not the derivative of f at a.

Equations

slope f a b = (b - a)⁻¹ • (f b -ᵥ f a)

theorem slope_fun_def {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) :

slope f = λ (a b : k), (b - a)⁻¹ • (f b -ᵥ f a)

theorem slope_def_field {k : Type u_1} [field k] (f : k → k) (a b : k) :

slope f a b = (f b - f a) / (b - a)

theorem slope_fun_def_field {k : Type u_1} [field k] (f : k → k) (a : k) :

slope f a = λ (b : k), (f b - f a) / (b - a)

@[simp]

theorem slope_same {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a : k) :

slope f a a = 0

theorem slope_def_module {k : Type u_1} {E : Type u_2} [field k] [add_comm_group E] [module k E] (f : k → E) (a b : k) :

slope f a b = (b - a)⁻¹ • (f b - f a)

@[simp]

theorem sub_smul_slope {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b : k) :

(b - a) • slope f a b = f b -ᵥ f a

theorem sub_smul_slope_vadd {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b : k) :

(b - a) • slope f a b +ᵥ f a = f b

@[simp]

theorem slope_vadd_const {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → E) (c : PE) :

slope (λ (x : k), f x +ᵥ c) = slope f

@[simp]

theorem slope_sub_smul {k : Type u_1} {E : Type u_2} [field k] [add_comm_group E] [module k E] (f : k → E) {a b : k} (h : a ≠ b) :

slope (λ (x : k), (x - a) • f x) a b = f b

theorem eq_of_slope_eq_zero {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] {f : k → PE} {a b : k} (h : slope f a b = 0) :

f a = f b

theorem affine_map.slope_comp {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] {F : Type u_4} {PF : Type u_5} [add_comm_group F] [module k F] [add_torsor F PF] (f : PE →ᵃ[k] PF) (g : k → PE) (a b : k) :

slope (⇑f ∘ g) a b = ⇑(f.linear) (slope g a b)

theorem linear_map.slope_comp {k : Type u_1} {E : Type u_2} [field k] [add_comm_group E] [module k E] {F : Type u_3} [add_comm_group F] [module k F] (f : E →ₗ[k] F) (g : k → E) (a b : k) :

slope (⇑f ∘ g) a b = ⇑f (slope g a b)

theorem slope_comm {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b : k) :

slope f a b = slope f b a

theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b c : k) :

((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c

slope f a c is a linear combination of slope f a b and slope f b c. This version explicitly provides coefficients. If a ≠ c, then the sum of the coefficients is 1, so it is actually an affine combination, see line_map_slope_slope_sub_div_sub.

theorem line_map_slope_slope_sub_div_sub {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b c : k) (h : a ≠ c) :

⇑(affine_map.line_map (slope f a b) (slope f b c)) ((c - b) / (c - a)) = slope f a c

slope f a c is an affine combination of slope f a b and slope f b c. This version uses line_map to express this property.

theorem line_map_slope_line_map_slope_line_map {k : Type u_1} {E : Type u_2} {PE : Type u_3} [field k] [add_comm_group E] [module k E] [add_torsor E PE] (f : k → PE) (a b r : k) :

⇑(affine_map.line_map (slope f (⇑(affine_map.line_map a b) r) b) (slope f a (⇑(affine_map.line_map a b) r))) r = slope f a b

slope f a b is an affine combination of slope f a (line_map a b r) and slope f (line_map a b r) b. We use line_map to express this property.