Lecture on Tangent Planes and Normal Vectors

Introduction

• Transition from finding tangent lines in Calculus 1 to tangent planes in Calculus 3.
• Tangent Lines: 1D, touch a curve at one point.
• Tangent Planes: 2D, touch a surface at one point in 3D.
• Goal: Understand how to find a tangent plane to a surface at a point.

Understanding Tangent Planes

• Concept: A tangent plane to a surface touches it at exactly one point within a neighborhood.
• Requirement for Plane: Need a normal vector besides a point.
• Importance of finding and defining a normal vector to establish a tangent plane.

The Concept of Normal Vector

• Normal Vector: Perpendicular to the tangent plane.
• Tangent planes have infinite vectors, but the normal vector is unique.
• Entire lesson focuses on deriving the normal vector.

Calculating Tangent Planes in 3D

• Surface Equation: Denoted as (f(x, y) = z).
• Level Curve: Intersection of the surface with a plane (parallel to the XY plane) creates a curve called a level curve.
• Projection: Projecting the level curve onto the XY plane results in a 2D curve.

Level Surfaces and Curves

• Level curves are one dimension less than the surface.
• Example: If surface is 3D, level curve is 2D.
• Use vector functions to represent these curves.

Deriving the Normal Vector

• Convert level curves to vector functions.
• Apply the chain rule to find derivatives of vector functions.
• Derivative results in a tangent vector to the level curve.
• Dot Product: Zero indicates orthogonality between gradient and tangent vector.

The Gradient

• Gradient: Gives the direction of steepest ascent.
• Gradient vector is the normal to level curves and surfaces.
• Application: Useful in finding tangent planes and normal vectors.

Practical Examples

• Problem Solving: Create a function one dimension higher than the given curve or surface.
• Gradient Calculation: Use it to determine the normal vector for tangent planes.
• Function Notation: Convert given equations to functions of variables (e.g., (x^2 - y^2 = 16) becomes (f(x, y) = x^2 - y^2)).

Steps to Find Tangent Plane and Normal Line

1. Identify given surface or curve.
2. Construct a higher-dimensional function treating variables as independent.
3. Calculate the gradient of the constructed function.
4. Use the gradient to find the normal vector to the level surface.
5. Use point and normal vector to determine the tangent plane and normal line.

Summary

• Finding tangent planes involves understanding surfaces, level curves, and gradients.
• Gradient vectors are central to finding normals and tangent planes.
• Approach involves constructing higher-dimensional functions and using calculus to derive the necessary vectors for problem-solving.

Practical Note: Understanding and practicing the derivation and application of gradients in finding tangent planes and normal lines are crucial for success in advanced calculus and related fields.