Definition of Derivative for Vector Functions https://youtu.be/LAXaiYA5p1k

In this video I go over the definition of derivative for a vector function, which is very similar to the definition for real valued functions. The derivative of a vector function is defined as the limit as h approaches zero of the secant vector (difference of two vectors separated by incrementing the parameter t by h) divided h. As h approaches zero the secant vector approaches a vector that is parallel to the tangent vector to the curve.

#math #science #calculus #vectorfunctions #geogebra

Derivatives and Integrals of Vector Functions https://youtu.be/YT4QJczx8KI

In this video I go further into vector functions and this time show how to obtain their derivatives and integrals. The definition of derivative for a vector function is much the same as with a real valued function, but the difference is that we apply this definition to each component of the vector function. Likewise, integration is also similar to real valued functions but the integral of each component is solved. This will be the basis for future videos when I cover velocity and acceleration of planetary bodies so stay tuned for that!

#math #science #calculus #vectorfunctions #education

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