If C is a circle in the plane, and if f(x, y)is differentiable and is not constant when constrained to C, then there must be at least one point on C where $\operatorname { grad } f$ is perpendicular to C.

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This statement is an application of the gradient theorem for scalar fields, which states that the gradient of a function at a point is always perpendicular to the level curve (or surface) of the function at that point. Since a circle in the plane can be considered a level curve of some function, and given that $f(x, y)$ is differentiable and not constant on $C$, there must be at least one point on $C$ where the gradient of $f$, $\operatorname{grad} f$, is perpendicular to $C$. This is because the gradient points in the direction of the greatest rate of increase of the function, which must be perpendicular to the direction in which the function does not change (i.e., along the circle $C$).

If C Is a Circle in the Plane, and If grad⁡f\operatorname { grad } fgradf

If C Is a Circle in the Plane, and If $\operatorname { grad } f$