## Quick Reference

Let *P* be a point on a smooth surface. The tangent at *P* to any curve through *P* on the surface is called a tangent line at *P*, and the tangent lines are all perpendicular to a line through *P* called the normal to the surface at *P*. The tangent lines all lie in the plane through *P* perpendicular to this normal, and this plane is called the tangent plane at *P*. (A precise definition of ‘smooth’ cannot be given here. A tangent plane does *not* exist, however, at a point on the edge of a cube or at the vertex of a (double) cone, for example.)

At a point *P* on the surface, it may be that all the points near *P* (apart from *P* itself) lie on one side of the tangent plane at *P*. On the other hand, this may not be so—there may be some points close to *P* on one side of the tangent plane and some on the other. In this case, the tangent plane cuts the surface in two curves that intersect at *P*. This is what happens at a saddle-point.

*Subjects:*
Mathematics.

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