Tangent Planes

Let $f: \mathbb{R}^2 \to \mathbb{R}$. Tangent plane at a point $P(a,b)$ is given by: $$

$$z = f(a,b) + \nabla f(a,b) \cdot \big(x - a, y - b\big)$$

To a level surface

Let

• $F(x,y,z)$ is a 3-variable function
• $S$ is the level surface of $F$ at $F(x,y,z)=k$
• $P = (x_0,y_0,z_0)$ be a point on $S$

The equation of the tangent plane on $P$ at the surface $S$:

$$\nabla F \cdot (x-x_0,y-y_0,z-z_0) = 0$$