mes 3Speak Podcast

Vector Functions and Space Curves

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https://3speak.tv/watch?v=mes/omhbcxkw
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!

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

                    • Notes and playlists:

                    • Summary: https://inleo.io/threads/view/mes/re-leothreads-vbej3cyq
                    • Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GwdMY-E9_4LsmYmO-IxcoX
                    • Notes: https://peakd.com/hive-128780/@mes/omhbcxkw
                    • Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U
                    • PDF Notes: https://1drv.ms/b/c/88862ef47bcaf6cd/EftibeFLloBPlVZwO6-DkXsB2UmaWm_0IBBQ6U-TWkM83g .

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                      mes 3Speak PodcastBy Math Easy Solutions