YouTube - Summary - Playlist - Vector Functions playlist
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. I also show this geometrically as a tangent vector to a point on a 3D surface and the limit as the second vector between that point and another point on the surface approach each other. The differentiation rules, such as the chain rule and product rule of dot products and cross products, have their similar counterparts for vector functions as well. 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
Timestamps
Introduction: 0:00
Calculus Book Reference: 0:47
Calculus Book Chapter: 1:13
Topics to Cover: 2:20
Derivatives and Integrals of Vector Functions: 3:55
Derivatives of Vector Functions: 4:14
(a) Secant Vector: 5:34
(b) Tangent Vector: 8:24
GeoGebra Animation of the Secant and Tangent lines: https://www.geogebra.org/3d/qn5zkfvy 11:15
Unit Tangent Vector: 14:15
Theorem: Derivative of a Vector Function by Components: 14:55
Example 1: Derivative and Unit Tangent Vector: 22:50
Example 2: Sketch the Tangent Vector: 32:19
Example 3: Tangent Line to a Helix: 44:49
Second Derivative of a Vector Function: 57:01
Differentiation Rules for Vector Functions: 58:52
Proof of Formula 4: Derivative of Dot Product of Vector Functions: 1:04:33
Example 4: Derivative of Vector Function is Orthogonal to the Vector: 1:11:50
GeoGebra graph of a Tangent Vector to a Sphere: https://www.geogebra.org/3d/qjahrjrw 1:21:22
Integrals of Vector Functions: 1:25:55
Example 5: Integral of a Vector Function by Components: 1:33:36
Exercises
Exercise 1: Tangent Vector points in a Direction of Increasing t: 1:39:07
Exercise 2: Proof of Formula 1: Derivative of Addition of Vector Functions: 1:50:39
Exercise 3: Proof of Formulas 2 and 3: Derivative of Vector Function Multiplied by a Function: 1:56:16
Exercise 4: Proof of Formula 5: Derivative of a Cross Product of Vector Functions: 2:04:12
Solution 1: Derivative of Components: 2:04:56
Solution 2: Definition of Derivative of Vector Functions: 2:19:03
Exercise 5: Proof of Formula 6: Derivative of a Function within a Vector Function: 2:32:38
Outro: 2:37:20
