Mathematical Physics Vol 1

Chapter 7. Partial differential equations

354

Proof The condition is necessary . For the relation (7.69) to be a total differential, the condi tions (7.70) must be satisfied ∂ p ∂ y + q ∂ p ∂ z − ∂ q ∂ x + p ∂ q ∂ z = 0 . (7.73) Further, by differentiating the functions f and g by x and y , respectively, we obtain ∂ f ∂ x + ∂ f ∂ z p + ∂ f ∂ p ∂ p ∂ x + ∂ f ∂ q ∂ q ∂ x = 0 , (7.74)

∂ f ∂ y

∂ f ∂ z

∂ f ∂ p ∂ g ∂ p ∂ g ∂ p

∂ p ∂ y ∂ p ∂ x ∂ p ∂ y

∂ f ∂ q ∂ g ∂ q ∂ g ∂ q

∂ q ∂ y ∂ q ∂ x ∂ q ∂ y

p +

= 0 ,

(7.75)

+

+

∂ g ∂ x ∂ g ∂ y

∂ g ∂ z ∂ g ∂ z

p +

= 0 ,

(7.76)

+

+

p +

= 0 .

(7.77)

+

+

∂ g ∂ p

∂ g ∂ q

∂ f ∂ p

∂ f ∂ q

Multiplying (7.74) by

, (7.75) by

, (7.76) by −

, (7.77) by −

, and then

adding, yields

[ f , g ]+ D pq

∂ p ∂ z −

∂ q ∂ z

∂ p ∂ y

∂ q ∂ x

+ q

+ p

= 0 .

(7.78)

Given that, according to the condition (7.73), the second member of the sum is equal to zero, it follows that [ f , g ]= 0 , that is, the functions f and g are in involution. The condition is sufficient . If the functions f and g are in involution, then [ f , g ]= 0, and thus from (7.78) it follows that the condition (7.73) is satisfied. The task was to find the conditions that should be satisfied by the function g = c 1 , so that we can determine from the system f = 0 , g = c 1 the values of p and q , but so that the expression p d x + q d y − d z = 0 is the complete differential of some function v (d v = p d x + q d y − d z ). Based on the previous Theorem, we can see that the functions f and g must be in involution, i.e.

[ f , g ]= 0 .

Using the definition (7.71), this condition, in expanded form, comes down to ∂ f ∂ p ∂ g ∂ x + ∂ f ∂ q ∂ g ∂ y ∂ f ∂ p p + ∂ f ∂ q ∂ g ∂ z −

+

q

−

p

∂ g ∂ p −

q

∂ f ∂ x

∂ f ∂ z

∂ f ∂ y

∂ f ∂ z

∂ g ∂ q

= 0 .

(7.79)

+

+