Tangent Plane

Summary

The tangent plane is the best linear approximation to a smooth surface $z=f(x,y)$ at a point. For an implicitly defined surface $F(x,y,z)=0$ , the gradient $\mathrm{\nabla }F$ is normal to the tangent plane.

Prerequisites

Partial Derivatives, Differentiability of a Function

Formula

Graph $z=f(x,y)$

If $f$ is differentiable at $(x_0,y_0)$ and $z_0=f(x_0,y_0)$ , the tangent plane is

Level surface $F(x,y,z)=0$

If $\mathrm{\nabla }F(x_0,y_0,z_0)\ne \mathbf{0}$ , the tangent plane is

Conditions / Assumptions

• Differentiability of $f$ (or $C^1$ partials as a convenient sufficient condition).
• For implicit surfaces, $\mathrm{\nabla }F\ne \mathbf{0}$ at the point (regular point).

Worked Example

For $z=x^2+y^2$ at $(1,1,2)$ , $f_x=2x$ , $f_y=2y$ , so

Common Mistakes

• Using the plane equation without evaluating partials at the point of tangency.
• Dropping the $z$ -term when the surface is given implicitly.

Connections

• Differential of a Function, Implicit Differentiation, Directional Derivative

References

Tangent planes are developed in OpenStax Calculus Volume 3.^[1]