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Often in physics my textbook refers to a "surface of constant X" where $X$ is some quantity like temperature, pressure etc.

However, it goes on to refer to the gradient quantity $\partial_i X$, where $\partial_i$ denotes the derivative with respect to some spatial coordinate. I don't understand why this derivative is not just $=0$?

I think my confusion is related to the definition of a level surface e.g. this question, but having scoured the available answers I still do not quite understand.

user1887919
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  • Suppose the ideal gas law $PV=nRT$ Suppose that $T$ is fixed, then $P\sim \frac 1V$. Have a look at any thermodynamic diagram. – Claude Leibovici Feb 22 '22 at 14:03
  • Could you please expand on your comment @ClaudeLeibovici? – user1887919 Feb 22 '22 at 14:16
  • Suppose that surface of constant $X$ is smooth. Then surface has normal vector. Along tangent $dX/dl={\rm grad} X \cdot \vec{l} = 0$ because surface has constant $X$. So any tangent is perpendicular to ${\rm grad} X$. Also any tangent is perpendicular to normal. That's why normal is parallel to ${\rm grad} X$. – Ivan Kaznacheyeu Feb 22 '22 at 14:23
  • When some $\partial_i X=0$ then appropriate normal vector component is equal to zero. When all $\partial_i X$ are equal to zero it means that normal vector direction is not defined in this point. When all $\partial_i X$ are equal to zero in some region, then $X$ is constant in whole region. – Ivan Kaznacheyeu Feb 22 '22 at 14:26

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