Useful relations between partial derivatives - University of Rhode Island

Useful relations between partial derivatives [tln6]

Consider state variables x, y, z, w. Only two of the variables are independent.

x

y -1

#1

=

y z

x z

x y z

#2

= -1

y z z x x y

x

x y

#3

=

w z

y z w z

x

x

x w

#4

=

+

y z

y w

w y y z

Proof:

x

x

Start with total differential of x(y, z): dx =

dy +

dz.

y z

z y

y

y

Substitute total differential of y(x, z): dy =

dx +

dz.

x z

z x

x y

x y

x

- 1 dx +

+

dz = 0.

y z x z

y z z x z y

Expressions in brackets must vanish independently #1 and #2.

Introduce parameter w: x(y, z) with y = y(w) and z = z(w).

x

x

dx x dy

dx =

dy +

dz =

+

y z

z y

dw y z dw

Specify path: z =const i.e. dz = 0: #3.

x dz .

z y dw

Introduce parameter y: x(y, w) with w = w(y) and z = z(y).

x

x

dx x

dx =

dy +

dw =

+

y w

w y

dy

y w

Specify path: z =const i.e. dz = 0: #4.

x dw .

w y dy

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