Tangent Planes and Normal Lines

Gradient of Functions with Two Variables

Let $f(x,y)$ be a function of two variables. Then, $\boldsymbol{\nabla}f$ is orthogonal to the level curve at point $P$ .

Gradient of Functions with Three Variables

Let $f(x,y,z)$ be a function of three variables. Then, $\boldsymbol{\nabla}f$ is normal (perpendicular) to the level surface at point $P$ .

Definition of a Tangent Plane

Let $P(x_0,y_0,z_0)$ be a point on the graph of $f(x,y,z) = c$ , where $\boldsymbol{\nabla} f \neq 0$ . The plane through $P$ that is normal to $\boldsymbol{\nabla} f$ evaluated at $P$ is called a tangent plane. The equation of such tangent plane is

(1) $\begin{equation*}f_x (x_0,y_0,z_0) (x - x_0) + f_y (x_0,y_0,z_0) (y - y_0) + f_z (x_0,y_0,z_0) (z - z_0) = 0.\end{equation*}$

Definition of a Normal Line

Let $P(x_0,y_0,z_0)$ be a point on the graph of $f(x,y,z) = c$ , where $\boldsymbol{\nabla} f \neq 0$ . The line that is parallel to $\boldsymbol{\nabla}f (x_0,y_0,z_0)$ and contains $P(x_0,y_0,z_0)$ is called the normal line to the surface at point $P$ . This line is given by

(2) $\begin{equation*}\frac{x-x_0}{f_x(x_0,y_0,z_0)} = \frac{y-y_0}{f_y(x_0,y_0,z_0)} = \frac{z-z_0}{f_z(x_0,y_0,z_0)} .\end{equation*}$

Reference

Dennis G. Zill. Advanced Engineering Mathematics, $6^{th}$ edition. Jones $&$ Bartlett Learning. 2016.