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**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 can

**r**(t) be a curve? I thought the position vector was a straight vector that stems from the origin of the coordinate system. My textbook shows

**r**(t) as a curved double arrow that lies on the surface in space.

Any help is appreciated.