Exercise 4: Parametric Equations of a Trochoid https://youtu.be/AbC3V1M7Voo

In this video I derive the parametric equations of a trochoid, which is the shape obtained by tracing a point as a circle rotates. If the point traced is on the circle itself, then the shape is the famous cycloid.

#math #trochoid #cycloid #desmos #calculus

Exercise 3: Manually Graphing the Projection of the Trefoil Knot onto the xy-Plane https://youtu.be/2_Zp2N35lYQ

In this video I manually (and painstakingly) plot an approximation of the projection of the 3D trefoil knot onto the xy-plane. I also plot the trefoil knot in 3D using the amazing GeoGebra graphing calculator as well as the new Desmos' graphing calculator. 

#math #trefoilknot #geogebra #desmos #calculus

Exercise 2: Properties of Vector Function Limits https://youtu.be/y5e8Xhdt39Q 

In this video I prove the main properties of vector function limits, namely the addition, constant multiplication, as well as the dot and cross products of their limits.

#math #vectorfunctions #limits #science #calculus

Exercise 1: Precise Definition of a Vector Function Limit https://youtu.be/mVJ4wviwDs8

In this video I go over the precise definition of a vector function limit and prove it in 2 different ways. The 1st solution involves assuming the vector exists, and then using the basic definition of a vector function limit as well as the 3D distance formula, to prove the difference inequalities of the precise definition. The 2nd solution is the reverse: it involves assuming the precise definition difference inequalities and then working backwards to prove the vector function limit exists. Lastly, I show that the precise definition of a vector function limit in 3D can be visualized as a sphere which gets indefinitely smaller as the interval shrinks.

This is probably the only in-depth precise definition of limits for vector functions anywhere in the internet, so buckle up!

#math #vectorfunctions #limits #science #calculus

Charged Particles Moving in Electromagnetic Fields https://youtu.be/48quQVxKZQ8

In this video I show that space curves can be used to model charged particle motion in orthogonally oriented electric and magnetic fields. Depending on the speed and position of a charged particle, the motion is either a cycloid or a trochoid (a more general version of a cycloid). I then take a detour into the mainstream physics of charged particle motion in electromagnetic fields in general, and read up on magnetic mirrors, null lines, Earth's magnetic field lines, Van Allen Belts, and the Northern Lights. Buckle up!

#math #nuclear #electromagnetism #GeoGebra #physics

Example 7: Using Computers (GeoGebra) to Draw Space Curves https://youtu.be/aBuDScugCXA

In this video I graph out several complicated space curves using GeoGebra's amazing 3D graphing calculator. These curves are the toroidal spiral, trefoil knot, and a twisted cubic.

#math #3D #spacecurves #GeoGebra #calculator