Vector Functions and Space Curves https://youtu.be/Qi2hVnKN-5c - Notes - Summary - Playlist - Vector Functions playlist

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)).

#math #science #education #vectorfunctions #calculus

Introduction to Vector Functions https://youtu.be/O7bGX857LSg - Notes - Summary - Playlist - Vector Functions playlist

In this video I introduce vector functions and show how they differ from the real-value functions that we are used to. Vector functions have domain of real numbers and a range of vectors. Compare this with real-valued functions which have both domain and range being single values, aka numbers.

#math #science #vectorfunctions #education #calculus

Limit of a Vector Function https://youtu.be/z48K6YRJ3Jc - Summary - Notes - Playlist - Vector Functions playlist

In this video I go over the definition of the limit of a vector function, which involves taking the limit of each of its component functions. If the limit exists, then so does the limit of each component function.

#math #limits #vectorfunctions #education #calculus

Space Curves: Tracing out the Tip of a Continuous Vector Function https://youtu.be/DHJnsmyG1LQ - Summary - Notes - Playlist - Vector Functions playlist

In this video I show that a continuous vector function in 3D forms a curve in space when tracing out the point on the tip of the vector as it moves in time.

#math #vectorfunctions #spacecurves #education #calculus

Plane Curves as Vector Functions and the Structure of a DNA Helix https://youtu.be/S94nUYS8Imc - Notes - Playlist - Vector Functions playlist

In this video I discuss how plane curves can be represented as vector functions, and illustrate an example of a plane curve that spirals upwards into a helix.

#math #vectorfunctions #planecurves #biology #calculus

Example 5: Vector and Parametric Equations of a Line Segment https://youtu.be/dSIfPYi3q4U - Summary - Notes - Playlist - Vector Functions playlist

In this video I go over an example in obtaining the vector and parametric equations of a line segment between two given points. I first recap on the vector equation of a line segment, which involves a parameter t between 0 and 1.

#math #vectorfunctions #geogebra #science #calculus

Exercise 1: Precise Definition of a Vector Function Limit https://youtu.be/mVJ4wviwDs8 - Summary - Notes - Playlist - Vector Functions playlist

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

Exercise 2: Properties of Vector Function Limits https://youtu.be/y5e8Xhdt39Q - Summary - Notes - Playlist - Vector Functions playlist

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

Derivatives and Integrals of Vector Functions - YouTube - Notes - 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. 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

Definition of Derivative for Vector Functions - YouTube - Summary - Notes - Playlist - Vector Functions playlist

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 by 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

Derivative for Vector Functions by Components + Proof - YouTube - Summary - Notes - Playlist - Vector Functions playlist

In this video I show that the derivative of a vector function is equal to the derivative of the real-valued functions in its components. I prove this by applying the definition of derivatives for vector functions, and then apply basic limit and vector laws to obtain the corresponding definition of derivative of each real-valued function in each component.

#math #science #calculus #vectorfunctions #proof

Vector Functions: Problems Plus - Notes - YouTube - Playlist - Vector Functions - mes.fm/math

In this video, I go over the end-of-chapter Problems Plus questions and solve 7 very advanced problems on vector functions. Three of the questions involve launching a projectile either on a flat, inclined, or declined plane. One question is on curvature, one on finding the equation of a plane, another is on rolling a ball off a table, and my favorite one is on finding the axial distance covered when wrapping a cable of known length around a spool of known diameter. Buckle up for 5 hours of mathematics!

#math #calculus #vectorfunctions #problemsplus #education

Problems Plus 1: Projectile fired from the origin - Notes - YouTube - Playlist - Vector Functions - mes.fm/math

In this video, I examine projectile motion launched from the origin and show that the projectile will always lie within a parabola defined by launching a projectile from the maximum height horizontally at the same initial velocity. I also show that if we are to aim directly at a target downrange and drop the target at the same instance of firing the projectile, then we will always hit our target!

#math #calculus #vectorfunctions #problemsplus #projectilemotion

Projectile aimed at a target downrange will always hit when dropped and fired at the same time! - Notes - Problems Plus - mes.fm/math

In this video, I show that firing a projectile while aiming at a distance target downrange and dropping it at the same time will always cause the projectile to hit the target, provided that the projectile doesn't hit the ground before reaching the target. This is because both the projectile and object accelerate downwards at the same rate due to gravity, and this is represented in the (-1/2)gt² term of the y-component in the parametric equations of projectile motion. This result is pretty amazing, and allows for some pretty cool stunts and trick shots to attempt!

#math #calculus #vectorfunctions #problemsplus #projectilemotion

Problems Plus 2: Firing a Projectile Up and Down an Inclined Plane - Notes - Problems Plus - mes.fm/math

In this video, I go over projectile motion along an inclined plane and show that the launch angle that gives the maximum range upwards or downwards is half the angle between the plane and the vertical. Even when firing at a target up an inclined plane, the angle that minimizes the energy used is also half the angle between the plane and the vertical. This result matches for a flat plane, since the maximum distance is achieved for a launch angle of 45°, i.e. half the angle between the plane (0°) and the vertical (90°).

#math #calculus #projectilemotion #vectors #vectorfunctions

Problems Plus 3: Physics of a Ball Rolling Off a Table - Notes - Problems Plus - mes.fm/math

In this video, I go over the physics of a ball rolling off a table and determine how far it falls, the angle and speed it bounces with, and the distance it travels for a second bounce after losing 80% of its energy upon impact.

#math #calculus #projectilemotion #vectors #vectorfunctions

Problems Plus 4: Curvature from Parametric Equations that involve Integrals - Notes - Problems Plus - mes.fm/math

In this video, I show how to obtain the curvature of parametric equations that involve integrals, by first utilizing the fundamental theorem of calculus to obtain the derivative of the position vector from the integrands of the integrals. Then we can follow the usual steps to obtain the curvature, which is defined as the magnitude of the unit tangent vector divided by the magnitude of the derivative of the position vector.

#math #calculus #curvature #vectors #vectorfunctions