crocomut said:Suppose we have a function
[tex]u=f(x,y,z)[/tex]
If [tex]\frac{\partial u}{\partial x} = 0[/tex]
then u is independent of x and is
[tex]u=f(y,z)[/tex]
only.
Correct?
gb7nash said:Correct. Think about it. If [itex]\frac{\partial u}{\partial x} = 0[/itex], this means that the value of u does not change whenever x changes. i.e. u does not depend on x.
When a partial derivative equals zero, it means that the rate of change of a multivariable function in a specific direction is equal to zero. This indicates that the function is not changing in that direction.
No, a partial derivative of zero does not always mean the function is constant. It only means that the function is not changing in a specific direction. The function may still be changing in other directions, resulting in a non-constant function.
Yes, a function can have a constant partial derivative but still not be constant. This is because the function may be changing at a constant rate in one direction, but changing at a different rate in another direction. Therefore, the overall function is not constant.
Partial derivatives can help determine constant functions by showing where the function is not changing. If all partial derivatives of a multivariable function are equal to zero, then the function is constant in all directions.
Yes, there are other conditions besides a partial derivative of zero that indicate a function is constant. These include having a constant value for all inputs, having a slope of zero at all points on the function, and having a constant rate of change in all directions.