Let w = f (x, y,z) be a differentiable function of x, y, and z. For example, suppose
Chapter 2, Problem 44(choose chapter or problem)
Let w = f (x, y,z) be a differentiable function of x, y, and z. For example, suppose that w = x + 2y + z. Regarding the variables x, y, and z as independent, we have w/x = 1 and w/y = 2. But now suppose that z = x y. Then x, y, and z are not all independent and, by substitution, we have that w = x + 2y + x y so that w/x = 1 + y and w/y = 2 + x. To overcome the apparent ambiguity in the notation for partial derivatives, it is customary to indicate the complete set of independent variables by writing additional subscripts beside the partial derivative. Thus, w x y,z would signify the partial derivative of w with respect to x, while holding both y and z constant. Hence, x, y, and z are the complete set of independent variables in this case. On the other hand, we would use (w/x)y to indicate that x and y alone are the independent variables. In the case that w = x + 2y + z, this notation gives w x y,z = 1, w y x,z = 2, and w z x,y = 1. If z = x y, then we also have w x y = 1 + y, and w y x = 2 + x. In this way, the ambiguity of notation can be avoided. Use this notation in Exercises 3945.The ideal gas law PV = kT , where k is a constant, relates the pressure P, temperature T , and volume V of a gas. Verify the result of Exercise 43 for the ideal gas law equation.
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