The document discusses conditions for integrability of an equation of the form X. dr = 0, where X is a vector function of x, y, and z. It shows that for the equation to be integrable, the curl of X must equal zero. This occurs when the partial derivative of Q with respect to x equals the partial derivative of P with respect to y. The solution of the integrable equation is then given as the logarithm of a ratio being set equal to a constant C.

1. Part II

2. If we express this equation in the form 𝑿. 𝑑𝒓 = 𝟎, then 𝑿 =
(𝑷 𝒙, 𝒚 , 𝑸 𝒙, 𝒚 , 𝑹 𝒛 ).

3. Then 𝐶𝑢𝑟𝑙 𝑿 =
𝒊 𝒋 𝒌
!
!"
!
!#
!
!$
𝑃(𝑥, 𝑦) 𝑄(𝑥, 𝑦) 𝑅(𝑧)
= 𝒊
𝜕𝑅
𝜕𝑦
−
𝜕𝑄
𝜕𝑧
+ 𝒋
𝜕𝑃
𝜕𝑧
−
𝜕𝑅
𝜕𝑥
+ 𝒌
𝜕𝑄
𝜕𝑥
−
𝜕𝑃
𝜕𝑦
= 𝒊 0 − 0 + 𝒋 0 − 0 + 𝒌
𝜕𝑄
𝜕𝑥
−
𝜕𝑃
𝜕𝑦
= 𝒌
𝜕𝑄
𝜕𝑥
−
𝜕𝑃
𝜕𝑦
= 0,0,
!%
!"
−
!&
!#

4. Hence, if the equa9on is integrable, i.e., if
We will have
!%
!"
−
!&
!#
. 𝑅 𝑧 = 0 ⇒
!%
!"
−
!&
!#
= 0 ⇒
!%
!"
=
!&
!#

10. Or
So, the solution is 2 log 𝑥 + log 𝑢 − log 1 + 𝑢 + 𝑣 = log 𝐶
i.e., log x'
+ log u − log (1 + u + v)=log C
⇒ log
"!(
)*(*+
= log 𝐶 ⇒
"!(
)*(*+
= 𝐶
10