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## Main Question or Discussion Point

I'm trying to understand why the gradient vector is always normal to a surface in space. My textbook describes

My question is, how can

Any help is appreciated.

**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.