Constrained variables illustration

In this example, consider

as a function of

, and

. In particular, let

. However, suppose

, and

have a relation among themselves. Specifically, suppose

. Assume that

is an independent variable.

If is independent, then depends on and , and (ultimately) is a function of and .
If is independent, then depends on and , and (ultimately) is a function of and .

So here is the question: Is

zero or nonzero? Surprisingly, the answer depends on whether

is chosen as the other independent variable. Use the illustration below to help you deduce the answer with geometric reasoning.

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