Partial derivative equals zero means it is constant?

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SUMMARY

The discussion confirms that if the partial derivative of a function u=f(x,y,z) with respect to x is zero, then u is independent of x and can be expressed as u=f(y,z). This indicates that changes in x do not affect the value of u. The conversation also highlights the nuanced distinction between "u does not depend on x" and "u is not a function of x," emphasizing the importance of precise terminology in mathematical discussions.

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crocomut
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Suppose we have a function

u=f(x,y,z)

If \frac{\partial u}{\partial x} = 0

then u is independent of x and is
u=f(y,z)
only.

Correct?
 
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crocomut said:
Suppose we have a function

u=f(x,y,z)

If \frac{\partial u}{\partial x} = 0

then u is independent of x and is
u=f(y,z)
only.

Correct?

Correct. Think about it. If \frac{\partial u}{\partial x} = 0, this means that the value of u does not change whenever x changes. i.e. u does not depend on x.
 
gb7nash said:
Correct. Think about it. If \frac{\partial u}{\partial x} = 0, this means that the value of u does not change whenever x changes. i.e. u does not depend on x.


It's interesting to contemplate the distinction between saying "u does not depend on x" and "u is not a function of x". For example, in the case of a single variable, the function that maps all real numbers to 3, which we write as f(x) = 3, is a constant function. But some people say it is "not a function of x" when they mean it "does not depend on x".
 

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