Important Mathematical Relations

Suppose there exists a relationship among the there variables and z. Then the following relation can be obtained.

• The chain rule of partial differentiation

(32.4)

• Reciprocal relation,

(32.5)

• Cyclic relation,

(32.6)

Jacobian Methods

(32.7)

In general can be written as

Let

This is a special case

(32.8)

Similarly

(32.9)

1. Properties:

(32.10)

2. Transposition

(32.11)

3. Inversion

(32.12)

4. Chain rule

(32.13)

5. Cyclic rule

	(32.14)
	(32.15)
	(32.16)

or

(32.17)

Let us consider

(32.18)

(32.19)

In Jacobian form

(32.20)

again

Let us consider

	(32.21)

Earlier we have obtained the Tds equations. These can be rewritten as

	(32.22)
	(32.23)

In addition, we know the availability function and Gibbs function

	(32.24)
	(32.25)

These equations can be generated through Mnemonic diagram (figure-32.1)

Figure 32.1

Differential (property) = differential of (independent variables)

Sign convention: going away from independent variable positive

Coming in toward independent variable negative

Equations (32.23)−(32.25) can now be expressed in the form

Let b be the dummy variable

	(32.26)
	(32.27)
	(32.28)
	(32.29)