Vector Functions and Space Curves

Math Easy Solutions

math science education vector functions calculus

description

In this video I deep dive nearly 5 hours into vector functions and space curves. A vector function has a domain (or input) of real numbers and a range (or outputs) of vectors. Compare this with the real-valued functions that we are used to, which have inputs of real numbers (typically x) and the outputs are also real numbers (typically y = f(x)).

Vector functions can be written as there component parts, which are made up of real-valued functions. In this chapter I focus on vector functions with 3 component parts because they can be plotted in three dimensional parts. Obtaining the parametric equations of such vector functions obtains a space curve. That is, a curve that can be plotted in 3D space. I also dive into the limits of vector functions, and show how to plot space curves with 3D graphing calculators such as GeoGebra. Buckle up!

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Timestamps:

Introduction: 0:00
Calculus Book Reference: 0:48
Calculus Book Chapter: 1:08
Topics to Cover: 1:40
Vector Functions: 2:37
Vector Functions and Space Curves: 4:04
- Example 1: 8:03
Limit of a Vector Function: Definition 1: 13:08
- Example 2: 17:04
Space Curves: 23:33
- Example 3: 29:07
Plane Curves: 33:15
- Example 4: 36:00
- DNA Helix: 44:03
Example 5: 48:10
Example 6: 1:00:20
Using Computers to Draw Space Curves: 1:19:21
- Example 7: Twisted Cubic: 1:22:37
- Visualizing Space Curves on Surfaces: 1:27:59
Charged Particle Moving in an Electromagnetic Field: 1:32:40
- The Physics of Charged Particles in Electromagnetic Fields: 1:44:32
Exercises
Exercise 1: Precise Definition of a Vector Function Limit: 2:14:20
- Recap on Real-Valued Function Limits: 2:15:48
- Solution 1: 2:19:40
- Solution 2: 2:36:54
- Visualizing the Precise Definition of a Vector Function Limit: 2:46:14
Exercise 2: Properties of Vector Function Limits: 2:54:25
- Solution: 2:55:44
- Solution to (a): 2:57:03
- Solution to (b): Constant multiple limit: 3:06:21
- Solution to (c): Dot Product limit: 3:10:35
- Solution to (d): Cross Product limit: 3:17:11
Exercise 3: Trefoil Knot: 3:28:51
Exercise 4: Trochoid: 4:21:00
Outro: 4:45:23