Let w = f (x, y, z) be a differentiable function of x, y, and z. For example, su...

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 = xy. Then x, y, and z are not all independent and, by substitution, we have that w = x + 2y + xy 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,

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,wewould 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

If z = xy, then we also have

In this way, the ambiguity of notation can be avoided. Use this notation in Exercises 39–45.

Verify the result of Exercise 43 for the ellipsoid ax² + by² + cz² = d where a, b, c, and d are constants.