Chain rule T beta
Temperature.chain_rule_T_beta
Plain-language statement
Chain rule for β(T) : d/dT F(β(T)) = F'(β(T)) * (-1 / (kB * T^2)), within Ioi 0.
Exact Lean statement
lemma chain_rule_T_beta {F : ℝ → ℝ} {F' : ℝ}
(T : Temperature) (hT_pos : 0 < T.val)
(hF_deriv : HasDerivWithinAt F F' (Set.Ioi 0) (T.β : ℝ)) :
HasDerivWithinAt (fun t : ℝ => F (betaFromReal t))
(F' * (-1 / (kB * (T.val : ℝ)^2))) (Set.Ioi 0) (T.val : ℝ)Formal artifact
Lean source
lemma chain_rule_T_beta {F : ℝ → ℝ} {F' : ℝ} (T : Temperature) (hT_pos : 0 < T.val) (hF_deriv : HasDerivWithinAt F F' (Set.Ioi 0) (T.β : ℝ)) : HasDerivWithinAt (fun t : ℝ => F (betaFromReal t)) (F' * (-1 / (kB * (T.val : ℝ)^2))) (Set.Ioi 0) (T.val : ℝ) := by have h_map : Set.MapsTo betaFromReal (Set.Ioi 0) (Set.Ioi 0) := by intro t ht show 0 < betaFromReal t rw [beta_fun_T_eq_on_Ioi ht] exact one_div_pos.mpr (mul_pos kB_pos ht) have h_beta_at_T : betaFromReal (T.val : ℝ) = (T.β : ℝ) := by rw [beta_fun_T_eq_on_Ioi (show (T.val : ℝ) ∈ Set.Ioi 0 from hT_pos), β_toReal] rfl have hF_deriv' : HasDerivWithinAt F F' (Set.Ioi 0) (betaFromReal (T.val : ℝ)) := by rw [h_beta_at_T] exact hF_deriv exact hF_deriv'.comp (T.val : ℝ) (deriv_beta_wrt_T (T := T) hT_pos) h_map- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Thermodynamics/Temperature/Basic.lean:217-233
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