I'm trying to understand why the gradient vector is always normal to a surface in space. My textbook describes(adsbygoogle = window.adsbygoogle || []).push({}); r(t) as a curve along the surface in space. Subsequently,r'(t) is tanget to this curve and perpendicular to the gradient vector at some point P, which implies the gradient vector to be a normal vector.

My question is, how canr(t) be a curve? I thought the position vector was a straight vector that stems from the origin of the coordinate system. My textbook showsr(t) as a curved double arrow that lies on the surface in space.

Any help is appreciated.

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# Gradient vector as Normal vector

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