The Geometric Applications of Differential Calculus

The conclusions of the previous chapters now have to land in geometry: the total differential gives us the tangent plane of a surface, and the directional derivative tells us how steeply a surface tilts in different directions.

The tangent plane and the normal line of a surface are completely determined by the gradient — the gradient is the normal vector, and that is more intuitive than any computed result. The tangent line and the normal plane of a space curve, on the other hand, come from differentiating the parametric equations, and their derivation is exactly the mirror image of the case of a surface: one requires “turning the derivative into a vector”, the other requires “eliminating a component”. Once you understand this difference, you will not confuse the two sets of conclusions.

By the end of this chapter, all the earlier abstract conclusions have become geometric figures that you can draw.