Mathematical Physics Vol 1
Chapter 7. Partial differential equations
348
that is
∂ ∂ x
∂ ∂ y
∂ ∂ y
∂ ∂ z
∂ ∂ z
∂ ∂ x
( vQ )=
( vP ) ,
( vR )=
( vQ ) ,
( vP )=
( vR ) ,
from where we obtain
v v v
∂ P ∂ y ∂ Q ∂ z ∂ R ∂ x
∂ v ∂ x
∂ Q ∂ x
∂ P ∂ y
∂ v ∂ y ⇒ ∂ v ∂ z ⇒ ∂ v ∂ x ⇒
∂ Q ∂ x − ∂ R ∂ y − ∂ P ∂ z −
∂ v ∂ y − ∂ v ∂ z −
∂ v ∂ x ∂ v ∂ y
Q
+ v
= v
+ P
= P
Q
,
∂ v ∂ y
∂ R ∂ y
∂ Q ∂ z
R
+ v
= v
+ Q
= Q
R
,
∂ v ∂ z
∂ P ∂ z
∂ R ∂ x
∂ v ∂ x −
∂ v ∂ z
P
+ v
= v
+ R
= R
P
.
These equations are linear partial equations of the first order. As we have shown earlier, the following systems of ordinary differential equations can be unambiguously assigned to them, and thus, from the first equation we obtain
∂ Q ∂ x −
∂ P ∂ y
∂ Q ∂ x −
∂ P ∂ y
d x − Q
d y P
d v
d v v
d x =
d y .
(7.43)
v
∂ P ∂ y
⇒
=
=
=
∂ Q ∂ x −
− Q
P
Similarly, from the second equation we obtain
∂ R ∂ y −
∂ Q ∂ z
∂ R ∂ y −
∂ Q ∂ z
d v v
d y =
d z ,
(7.44)
=
− R
Q
and from the third
∂ P ∂ z −
∂ R ∂ x
∂ P ∂ z −
∂ R ∂ x
d v v
= (7.45) Given that the function on the left hand side of these equations is the same, it follows that the coefficients next to the corresponding differentials are equal, i.e. R d x = − P d z .
∂ Q ∂ x
∂ P ∂ y
∂ P ∂ z −
∂ R ∂ x
−
+
=
,
Q
R
∂ R ∂ y −
∂ Q ∂ x −
∂ P ∂ y
∂ Q ∂ z
=
,
P
− R
∂ R ∂ y −
∂ Q ∂ z
∂ P ∂ z −
∂ R ∂ x
=
.
Q
− P
From here we obtain
Q R P
∂ R ∂ x ∂ P ∂ y ∂ Q ∂ z
= R = P = Q
∂ Q ∂ x ∂ R ∂ y ∂ P ∂ z
∂ P ∂ z − ∂ Q ∂ x − ∂ R ∂ y −
∂ P ∂ y − ∂ Q ∂ z − ∂ R ∂ x −
,
,
.
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