In the previous set of supplemental notes, when we were finding partial derivatives of functions like w = f (x, y), we have assumed that x and y were independent variables. This is not always the case. Suppose the variables in a function w = f (x, y, z) are constrained by a relation imposed on x, y, and z. Then the partial derivatives of f will depend on what variables are chosen to be dependent and which are chosen to be independent.

EXAMPLE 1:

SOLUTION:

independent variables, and w and t to be the dependent variables. It is sometimes useful to draw an arrow diagram to show how the variables and functions are related.

Independent variables

Intermediate variables and relations

x = x, y = y, z = z,

t = x + y

Dependent variable

EXAMPLE 2:

SOLUTION:

Independent variable

Intermediate variables and relations

y = y, z = z, t = t,

x = t - y

Dependent variable

EXAMPLE 3:

Let U = f (P, V, T) be the internal energy of gas that obeys the ideal gas law PV = nRT (n and R constant).

SOLUTION:

Independent variables

Intermediate variables

P = P, V = V, T = (PV)/(nR)

Dependent variable