Tangent Planes and Linear Approximation to Surfaces

In one variable, the tangent line at a point gives the best linear approximation to the curve. In two variables, the tangent plane at a point $(x_0,y_0,f(x_0,y_0))$ gives the best linear approximation to the surface $z = f(x,y)$. The tangent plane equation is the geometric version of the total differential: $z = f(P_0) + f_x(P_0)(x-x_0) + f_y(P_0)(y-y_0)$. In options pricing, this is the delta approximation — the tangent plane to the option price surface in $(S,\sigma)$-space is the first-order (delta, vega) approximation to the P&L. The normal vector to the surface is $\nabla F$ (for a level surface $\{F = c\}$), which also gives tangent planes to implicitly defined surfaces such as the yield-curve constraint $F(y,P) = 0$.

Learning objectives. By the end of this note you should be able to: