Partial derivative equals zero means it is constant?

[crocomut]

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?

 

[gb7nash]

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.

 

[Stephen Tashi]

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".