Section 14.2

Gradient Applications

Why is the key to surface geometry.

Introduction

In the last section, we found tangent planes for . But what if we have a surface defined implicitly like ?

The Gradient Vector gives us a powerful shortcut: it is always perpendicular (normal) to the level surface.

Gradients as Normals

Key Fact

If defines a surface, then the vector

is normal to the surface at every point.

Visualizing Normal Vectors

Interactive: Normal to a Sphere

Equations

Tangent Plane Equation

Using as the normal vector:

Normal Line Equation

Using as the direction vector:

Worked Examples

Example 1: Ellipsoid Tangent Plane

Find the tangent plane to at .

1. Define F:

2. Find Gradient:

At : .

3. Write Plane Equation:

Simplify: .

Example 2: Normal Line

Find the normal line to at .

1. Rewrite as Level Surface:

2. Gradient:

At : .

3. Line Equation:

Paramentric: .

Example 3: Angle Between Planes

Find the cosine of the angle between and at .

Angle between surfaces = Angle between their normal vectors.

1. Normals:

.
.

2. Dot Product and Magnitudes:

3. Cosine: