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