Tangent Plane

Tangent Planes to Level Surfaces

A level surface is defined by an equation of the form $F (x, y, z) = k$ , where $k$ is a constant. The surface represents all points $(x, y, z)$ where the function $F$ takes the value $k$ .

1. Gradient and Tangent Planes:

The gradient vector $\nabla F = ⟨ F_{x}, F_{y}, F_{z} ⟩$ at a point $(x_{0}, y_{0}, z_{0})$ on the surface $F (x, y, z) = k$ is crucial because it points in the direction of the greatest rate of change of $F$ . Moreover, the gradient vector is perpendicular (or normal) to the level surface at that point.

2. Equation of the Tangent Plane:

At a point $(x_{0}, y_{0}, z_{0})$ on the level surface $F (x, y, z) = k$ , the tangent plane to the surface can be found using the gradient vector. Since the gradient $\nabla F (x_{0}, y_{0}, z_{0})$ is normal to the surface, the equation of the tangent plane is:

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

This equation ensures that any point $(x, y, z)$ lying on the tangent plane satisfies the condition of being perpendicular to the gradient vector at $(x_{0}, y_{0}, z_{0})$ .

3. Example:

Suppose we have a level surface given by $F (x, y, z) = x^{2} + y^{2} + z^{2} - 9 = 0$ , which represents a sphere of radius 3. The gradient vector $\nabla F = ⟨ 2 x, 2 y, 2 z ⟩$ is normal to the surface at any point.

To find the tangent plane at the point $(2, 1, 2)$ , calculate the gradient at that point:

\nabla F (2, 1, 2) = ⟨ 4, 2, 4 ⟩

Thus, the equation of the tangent plane at $(2, 1, 2)$ is:

4 (x - 2) + 2 (y - 1) + 4 (z - 2) = 0

4 x + 2 y + 4 z = 18

Summary

The gradient vector of $F (x, y, z)$ at a point on a level surface $F (x, y, z) = k$ is normal to the surface and defines the tangent plane at that point.
The equation of the tangent plane is derived using the components of the gradient vector, resulting in a plane perpendicular to the gradient at that point.