In this example, consider [math]w[/math] as a function of [math]x[/math], [math]y[/math], and [math]z[/math]. In particular, let[br] [math]w=x^2+y^2+z^2[/math].[br]However, suppose [math]x[/math], [math]y[/math], and [math]z[/math] have a relation among themselves. Specifically, suppose[br] [math]z=x^2+y^2[/math].[br]Assume that [math]x[/math] is an independent variable.[br][list=1][*]If [math]y[/math] is independent, then [math]z[/math] depends on [math]x[/math] and [math]y[/math], and [math]w[/math] (ultimately) is a function of [math]x[/math] and [math]y[/math].[/*][*]If [math]z[/math] is independent, then [math]y^2[/math] depends on [math]x[/math] and [math]z[/math], and [math]w[/math] (ultimately) is a function of [math]x[/math] and [math]z[/math].[/*][/list]So here is the question: [i]Is[/i] [math]\frac{\partial w}{\partial x}[/math] [i]zero or nonzero[/i]?[br]Surprisingly, the answer depends on whether [math]y[/math] or [math]z[/math] is chosen as the other independent variable. Use the illustration below to help you deduce the answer with geometric reasoning.