The Adventur

es of Vincent

the Vector-

1

Valued Function

By Krystal Hoganson*

*does not own foxtrot comic, but http://foxtrot.com does;*did not draw all illustrations;*did use publicly available internet resources to compile this piece;*does not guarantee the complete accuracy of all formulas, statements, equations, or theories; *does asks that the work not be duplicated in any manner without permission or acknowledgement;*does has an email address khoganson@miners.utep.edu .

Chapter 1: Vincent’s Quest Begins with Vector-Valued Functions

In a book, on a shelf, in any given Calculus class, lives a very odd collection of functions. This unique family provides the ability for motion to be represented along a curve or the graph of a curve to be traced. They are the family of Vector-Valued functions and they map real numbers to vectors. That is their purpose, while they serve many different roles in everyday life.

One particular Vector-Valued function, we shall call him Vincent, has come of an age where it is time for him to go out into the mathematical world and serve the purpose of a crazed Calculus instructor. Will he represent a model hurricane swirling about an axis? Will he be used in a sports analysis where he inevitably is graphed in every which manner to determine the projected landing of a shot put after it has been put?

As you can imagine, Vincent has a great deal of questions for his future. With trepidation he approaches his father, also a Vector-Valued Function, to receive some answers to his questions.

“Well it is really quite simple, my boy,” his father begins with enthusiasm. “First it is imperative to understand that us vector-valued functions take real numbers as arguments to produce vectors as a simplified output.”

Vincent blinks, dumbfounded.

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With a sigh, his father offers some clarification. “Real numbers go in, vectors come out.” At this Vincent smiles, but that too soon wavers.

“But father, if it is really that simple, why are we considered so complex?” “Well, son, like other vectors we can exist, or be graphed, on a plane curve or a space curve,” his

father did not wait for the look of complete confusion to break Vincent’s face before he continued. “You see, a plane curve is defined by a set of ordered pairs and exists in two dimensions. These ordered pairs are continuous functions and can be defined by their respective parametric equations. Operating in three dimensions is quite similar, however, difficult for humans to envision without the aid of computers.”

Vincent thought on that for a moment. “So our complexity rests in the limited scope of the humans’ imagination?”

His father chuckled outright. “Not entirely, my boy. Vector-Valued functions existing on a space curve are especially difficult to represent through traditional methods. Our domain is the intersection of the domain of our respective parametric equations acting as individual functions, essentially becoming a function in itself. As you may have noticed, we are not solid but more of a collection of vectors, yet we are still functions. We do not pass the vertical line test, yet we are still functions. It is entirely too much for someone to understand without seeing a Vector-Valued function represented correctly. That is what makes us complex.”

“You keep saying ‘see’ or ‘graph’. Why is this so important?”“There are two roles that Vector-Valued functions serve in the representation of curves. We

either represent motion along a curve or trace the graph of a curve.”Vincent suddenly exclaimed. “Calculus! Calculus is the study of motion!”“Precisely. And with Calculus we can manipulate real numbers or vectors to find a function’s

velocity, acceleration, continuity, limit, curvature, and so forth. And do you know what?” At Vincent’s shake of his head, his father continued. “It is possible to perform all of those operations on a Vector-Valued function as well.”

“But,” Vincent began, “how will I know what to do, should it be asked of me?”“That is the best question you’ve asked all day, m’boy. In order to get out of this book, you need

to have that question answered. I propose that you forego making yourself available to that deranged Calculus instructor until you have all of your questions answered first. It would not be right to send you out onto the chalkboard and have you confusing those learning humans.”

“I think you are right, father. When will you be teaching me?” Vincent asked a smile on his face.“Whoa, hold on there. I have a date with an Archimedes Screw in a moment or two,” his father

said but after seeing his son’s crestfallen expression offered a suggestion. “Your Aunt Linda knows everything to know about taking the limit of a Vector-Valued function. I suggest you start with her and see where that knowledge takes you.” His father smiled and patted him on his head as he jumped off the pages and was promptly scrawled onto a blackboard by one of those frenzied Calculus instructors.

With a sigh Vincent steeled himself for a visit with his Aunt Linda. He hoped that he could share in his father’s confidence that he would grasp what was required of him, so as not to besmirch the ancestral line of functions that made up his extended family. He quickly glanced at the wall of family portraits before making his way over to see Aunt Linda at the end of Section 12.1.

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4

Chapter 2: Limits of Vector-Valued Functions

“I have been wondering when your father would send you to me, Vincey,” his Aunt Linda gushed, using her nickname for him. “Limits, limits, limits… where to start?” She asked herself.

“Well,” Vincent started, “maybe you can help me understand exactly what a limit is. I never can keep it all straight.” He suggested. It was the truth. He had gone this entire time, to the near end of the Calculus book, and still wasn’t quite sure what came before him.

“That is a good place to begin. A limit can be regarded in a number of different methods, but ultimately the limit of a function is the function itself trying to reach its intended height.”

“Errr.. intended what?” Vincent scratched his head, wondering if he was really ready for this.“Think of Archimedes, the Father of Mathematics. He assumed that PI was the circumference of

the circle divided by the diameter. He understood that if he were to draw a polygon inside the square, the more sides were added to the polygon the more it resembled a circle.”

will do my best to ensure that it becomes easy enough for you as well.” She began to shuffle through the rules and definitions below her section. “Right. First thing we must know is that if r is a vector valued function such that r(t)=f(t)i+g(t)j+h(t)k, then—“

“Wait! What is that?”His aunt blinked as if he had asked a very outlandish question.“That is our Vector Valued function…” She stated, finally understanding the lack of knowledge

Vincent possessed. He silently mouthed, “Oh.” and his aunt continued.“Now where was I? Oh, yes. If that statement for r is true then the limit as t approaches a is

equal to the limit of each of the components. It can be expressed as followed:

Vincent looked over the paper closely. “I understand what a limit is… but something tells me there is more.”

“Indeed there is, Vincey. You can manipulate the limits of vector-valued functions just as you would limits of functions. You can find the limit of the sum of two vector-valued functions. You can find the product, quotient, or power of the limit of vector-valued functions as readily as you can the limit of any other function.”

“Right. As long as it exists?” Vincent asked, some of the Calculus coming back to him.“Alright, smarty-pants, what determines if a limit exists?” His aunt quizzed him.

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His aunt quickly began scribbling shapes onto a piece of paper.

“As the polygon gets larger, or rather as the number of its sides increases, it gets closer and closer to being a circle. As the polygon’s sides approach infinity it approaches the circle. The limit of the polygon as its sides go to infinity is the circle!”

Vincent proclaimed with understanding, “So the polygon’s intentions are to be a circle. It will come close, but will never actually become one.”

“Exactly, Vincey. So in terms of finding the limit of Vector-Valued functions we must take the limit of each of the components.”

Vincent thought for a moment on this. “I assume it is not as easy as it sounds.”

His aunt smiled. “Well, easy enough for me, but I

“…magic?” He asked sheepishly.“Let’s try to be serious, Vincent.” She said sternly, her playfulness gone. “If the function

approaches a different number from the right than it does from the left, if it increases or decreases without bound or oscillates between two fixed values, in the polygon example I gave before, as the number of sides approaches infinity it can be assumed the limit does not exist.”

“And these rules apply to the limits of vector-valued functions as well?” Vincent asked, vaguely remembering these rules from Chapter 1.

“Of course. Remember, the limit of a vector valued function is the limit of each of the components of the vector-valued function,” Aunt Linda reminded him. “However, I am afraid that is the extent of my expertise.”

“What do you mean?”With a sympathetic smile she began to usher him to the next portion of section 12.1. “You

didn’t think that was it, did you? There is much you need to learn about Vector-Value functions before you can get out of this book. Speak with Connie next. She can tell you about Continuity of Vector-Valued functions.”

“What does taking the limit of a vector-valued function have to do with continuity?” Vincent asked the blank look washing over his face once again.

“They have as much in common as limits of functions do with continuity. The Properties of Continuity are built upon the properties of limits. If b is a real number and f and g are continuous at x=c, then the scalar multiple bf, the sum and difference f±g, product fg, and quotient f/g are also continuous. Does any of that sound familiar, Vincey?”

Vincent shook his head as if to clear his thoughts. Clearly, perplexed he responded, “Vaguely.”Linda, in turn, shook her head as well. “Well, I don’t have all the time in the world to sit here

explaining it to you, Vincey. You go find your Cousin Connie, tell her I sent you to find out all she knows about continuity.” At his look of dejection, she sighed. “Cheer up, Vincey. You’ll get out of this book and onto that board in no time.”

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Chapter 3: Continuity of Vector-Valued Functions“So. Linda sent you my way to learn about Continuity of Vector-Value Functions, eh?” His

Cousin Connie said after he had explained why he was thrust so unceremoniously into her section of the chapter.

“She did say you were the best,” Vincent replied. “Well that’s because I am!” She said with a twirl. “Now… Continuity. Let’s see. The given

definition of continuity of a VVF—mind if we shorten it?” She asked. At the shake of his head, she continued. “A VVF r is continuous at the point given by t=a if the limit of r(t) exists as t→ a and lim as t→ a of r(t)=r(a).”

Vincent sat with his mouth agape.Connie sighed. “A VVf is continuous at r(t) if it is defined at t.”Vincent closed his mouth. “And is continuity like taking the limit of a Vector-V… VVF? By

looking at each component?”Connie smiled. “Indeed it is.”Vincent realized she wasn’t going to say anymore. “That’s it? No weird analogies, pictures, or

frowning at me?”“Well no. You must understand that the VVF r is continuous on an interval I if it is continuous at

every point in the interval. From that VVF is continuous at t=a if and only if each of its component functions is continuous at t=a.”

Vincent rubbed his temples feeling a calculus migraine coming on.“Listen, Vincent. They don’t call me Continuity Connie for nothing. Just think of a continuity of a

VVF as being a function that is continuous at a certain point. Meaning there are no holes, jumps or gaps. You might say that a function is continuous an open interval if its graph can be drawn with a pencil without lifting the pencil from the paper.”

“But my father said that humans can’t draw vector-value functions to save their life! How will they know if a VVF is continuous on an open interval if they can’t even bring a pencil in to draw the function?”

“Humans can understand functions. They would just look at each component and not the whole function.”

Vincent nodded.“At this rate, I might just make it out of this book and on to a chalkboard sooner than I thought.”“Whoa hold on there. You still need to learn about taking the derivative and integrating a VVF.

You’re not out of the woods yet.”Vincent sighed knowing that this meant he was through with this section and was being pawned

off onto someone else. “Who do I go talk to now?”“Don’t sound so inconvenienced, Vincent. Calculus is the study of motion. It takes all

components, working together, to derive the answers. You need to understand that everything you’ll learn about VVF tie into everything else. You can’t have one without the other. On that note, go see Derivative Dan, your uncle on your mother’s side. He’ll set you straight in that arena.”

Vincent left her section to track down crazy Uncle Dan. By the time he knew everything he was supposed to know about himself, the entire family would have yelled at him at least once. This was going to be a long adventure, indeed.

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Chapter 4: Derivatives and Integration of Vector-Valued FunctionsUncle Dan was as ancient as they came. Vincent had heard at one point that his Uncle suffered

from Shellshock having been written on a board and erased more often than any other function in the textbook. That left Uncle Dan jumping at every noise, barricading himself against the enemy Freshmen, and more often than not up to a lot of trouble. Knowing this, Vincent approached him with great caution.

“Uncle Dan?” He called out with trepidation into the section.“IDENTIFY YOURSELF!” Came the terse response.“It’s me, Uncle Dan. Vincent.”All he got in response was a mutter “Hmph.”“Erm. Uncle Dan? I’m here to see if you can tell me what you know about taking the derivative

of vector-valued functions…?” Vincent said a bit more loudly as he started making his way into the section.

“Hold it there boy!” Vincent stopped immediately. His uncle quickly approached from the top of the page and stood before him. “Derivatives you say?”

“Yes, sir.”“Well you take them for vector-valued functions the same way you do any other function. Now

stop bothering me,” he said in a huff and continued on through the page. Vincent looked around warily, but continued after him.

“Yes, sir. But I’m in a bit of trouble. I—“ he was quickly cutoff.“Trouble you say?! Is it the darned Freshmen? I knew they’d come after my family. What have

they done to you boy? Speak up! This is a matter of life and death!” His uncle said, fiercely shaking his shoulders.

“No, no. Nothing like that. You see I want to get out of this book and onto the board. But, in order to do that, I need to know what my purpose in the world will be…. I was hoping you could help me with derivatives, sir.” He received another hmph in response but was not deterred. “You said that vector-valued functions are like other functions in terms of taking the derivative. Does that me one can differentiate a vector valued function by differentiating the component functions?”

“Your mother put you up to this?” His uncle asked looking at him questioningly. Vincent just shook his head and his uncle gave him another hmph. “The derivative is defined in terms of limits. Taking the limit of a vector valued functions amounts to taking the limits of the functions. Since the component functions are real-valued functions of one variable, all the properties of the derivative will hold. Where u⃗∧v⃗ are differentiable vector-valued function c=constnad and f=real-valued function the following statements are true: (u⃗(t )± v⃗ ( t)¿=u⃗ ' ± v⃗ ' (t) ,(c u⃗(t )) '=c u⃗' (t) ,( f (t ) u⃗(t )) '=f ' (t) u⃗(t)+ f (t ) u⃗' (t) ,¿(t))’=

u⃗' ( t )∗v⃗ (t )∗u⃗ ( t )∗v⃗ ' (t ) , (u⃗ ( t )× v⃗ (t ) )'= v⃗ (t)¿ '=u⃗' (t)× v⃗ ( t)+ u⃗(t)× v⃗ ' (t) ,(u⃗( f (t )))'=f ' ( t ) u⃗' ( f (t)) .” Vincent nearly fainted trying to remember all of that, let alone understand it.

“Okaaay. What does all that mean?”“Dang, boy! How’d you get this far into the book without retaining anything?” He asked shaking

his head in disgust. “They’re essentially the same rules for taking the derivative of a function, but in this instance vector-valued functions are the name of the game. I don’t have time to be sitting around here refreshing your memory, I’ve just got to move forward and assume you know what the heck I’m talkin about.” Vincent realized that he was going to have to continue on with his uncle and do the best he could.

“Okay, sir. So what else can you tell me about derivatives and vector-valued functions?”“Well, essentially, you could look at it like this:

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“And because the derivative of a sum is the sum of the derivative, we can find the derivative of each of the components of the vector valued function to find its derivative.” Uncle Dan finished gruffly.

“Right. Anything else?” Vincent asked feeling another calculus migraine coming on.His uncle snorted at him and scribbled something on a piece of paper and shoved it in front of

him.

Vincent looked it over a few moments, a light bulb finally going off.“So limits help in proving the properties of taking a derivative of a vector-valued function,” he

formulated aloud.“Finally, some progress, boy.” Uncle Dan said with more than a mild case of exasperation.

“Listen, boy. Derivatives of vector-valued functions are not complicated. The derivate r’ of a vector-valued function is defined in much the same way as for real-valued functions. The derivative is defined in terms of limits. Taking the limit of a vector valued function amounts to taking the limits of the component functions. Since the component functions are real-valued functions of one variable, allthe properties of the derivative will hold.”

“I’m starting to see that. Thanks Uncle Dan. I think I have all the information I need to escape this book.”

At that remark his uncle began to chuckle loudly.“You kiddin’ me? You still need to learn about integration, projectile motion, unit tangent

vectors, principal normal vectors, arc length of a space curve, the arc length function, curvature, and the formulas of curvature. You’re just getting started.” He slapped Vincent on the back and began to retreat back into his section.

“So who do I go see next?” Vincent called after him.Uncle Dan stopped for a moment and called over his shoulder, “Do you remember how to

integrate functions?”Vincent nodded. “Kind of.”“Well, I guess I can cover that one since I’ve got you here already.” Uncle Dan returned to

where Vincent hadn’t moved from. “The definite integral of a vector-valued function is defined in exactly the same way as the derivative of an ordinary function. You can define the integral of a vector-valued function as the integral of each component. This produces three different constants, one for each component. Those three scalar constants produce one vector constant of integration.”

“That’s it?”Uncle Dan smacked the back of his head.

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“You’re not listening to anything. Your generation wants everything to be handed to you. You don’t want to learn, because you’re lazy!” He made to hit Vincent again, but Vincent quickly dodged away. His uncle chuckled again. “But when you have to learn, you do so very quickly, don’t you boy?”

Vincent knew better than to answer that question.“With real-valued functions, you can narrow the integrals down to a single integral by setting an

initial condition. Now that I’ve told you everything you need to know, why don’t you get out of my section and head over to Peggy May. She’ll teach you what you need to learn about projectile motion.” He turned abruptly and went back into his section.

Vincent shook his head and wondered what else was in store for him on this adventure.

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Chapter 5: Projectile Motion“What goes up, must come down,” Peggy May began with a smile amidst the mystic ambience

she had created. Where she got the fog machine is still a mystery. “Once you understand that, you’ll understand the entire universe.”

“Except, when there’s no gravity,” Vincent muttered. Peggy May frowned and the fog dropped off the page quite suddenly.

“You want my help or not?”“Sorry. I don’t understand, though. What does vector-valued functions have to do with

projectile motion?” He asked.

“You said you spoke to Linda about Limits, Connie about Continuity, and your Uncle Dan about integrating and differentiating vector-valued functions. Well, projectile motion has to incorporate all of those elements in order to arrive at the correct answer.

“Velocity and acceleration are key. In single variable calculus the velocity is defined as the derivative of the position function. For vector-valued functions, we make the same definition. Let r(t) be a differentiable vector valued function representing the position vector of a particle at time t. Then the velocity vector is the derivative of the position vector.

v(t) = r'(t) = x'(t)i + y'(t)j + z'(t)k

“The same applies for acceleration. There is no change from single variable calculus to vector-valued functions. Let r(t) be a twice differentiable vector valued function representing the position vector of a particle at time t. Then the acceleration vector is the second derivative of the position vector.

a(t) = r''(t) = x''(t)i + y''(t)j + z''(t)k

“Since the velocity and acceleration vectors are defined as first and second derivatives of the position vector, we can get back to the position vector by integrating. When an object is propelled into the air, it is assumed that all other forces acting on the object except gravity are negligible. This includes any effects due to air resistance on the object. Essentially we have all we need to derive the parametric equations for the path of a projectile once we know that the motion occurs in a vertical plan represented by the xy coordinate system. The origin is a point on the earth’s surface. For a projectile of mass m, the force due to gravity is:

F=-mgj

Gravitational constant g is equal to 9.81 meter/second 2 or 32 feet per second. We can apply Newton’s Second Law of motion, which states, that the force applied to a body produces a proportional acceleration; the relationship between the two is

F=ma

where F is the force applied, m is the mass of the body, and a is the body's acceleration thus giving us

ma=-mgj

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“Ultimately giving us the final equation,

a=-gj

“And that’s projectile motion.” Peggy May finished looking smug.This was easily the most confused Vincent had been the entire adventure. He looked around to

see if this was a joke, but realizing that Peggy May was serious, he gulped, got an idea and replied, “Ok. So vector-valued functions follow the same rules in terms of single-variable calculus to derive the acceleration and velocity vectors by taking derivatives. Then after integrating you can arrive, with a constant, including height, speed, and the angle at which the projectile was launched. Then from the given height you can equate an additional vector while the speed gives the magnitude of the initial velocity of which you can use to equate the position vector.”

“Where did you come up with that last bit? I hadn’t even covered initial height, initial velocity, or position vector?” She asked getting suspicious. Vincent began looking guilty.

“I was reading the paragraph behind you. BUT,” he continued when Peggy May began glaring at him, “this is all very helpful. I think I understand it all a little bit more.”

“Well, we’re almost done here. Take a look at this masterpiece,” she said, directing his attention to a figure in the book. It looked like any other figure, expect Peggy May had altered it considerably.

“Between you and me kid, I think it looks better this way,” she beamed. “Each point on the path of motion of the projectile is represented by a vector in terms of height and velocity. Sir Newton would be proud.”

Vincent nodded his agreement. Peggy May pulled out another piece of paper and gave it to Vincent. “This is the equation to find the position vector. It will definitely come in handy.”

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r(t)=(v0cosθ)ti+[h+v0sinθ)t-1/2gt2]j

“And that is projectile motion.”Vincent nodded, finally tying everything together. Vector-valued functions were becoming more

and more clear to him. His father was right. His questions were being answered and he was confident that he would make it out of the calculus book after all.

“Who should I go speak to next?”“Your sister Ursula knows about Unit Tangent Vectors. Go see her. And good luck Vincent!”

She said ushering him into the next section. On his way out, he spied a comic framed on Peggy May’s section that made him smile.

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Chapter 6: Unit Tangent Vectors and Unit Normal Vectors“What do you want, twerp?” His sister asked him coldly. Growing up in a calculus section with

his sister hadn’t been fun for either one of them; sibling rivalry had decayed what semblance of civility they might have had at one point. Vincent needed her now and hoped that he could appear as charming as possible.

“I know you’re mad I threw away your equations, but that was pages ago! Will you accept my apology for my behavior?” He asked with a small smile and pleading eyes.

“Like I said, what do you want? You’ve never come to see me before…,” she asked again.“Well I came by to say I was sorry…?” He said unsure of himself all over again. She continued to

look at him disbelievingly. He was going to have to try a different a different route: flattery. “Listen, Ursula, we both know that you’re the smartest function in this book and you know the most when it comes to unit tangent vectors. I need your help answering a few questions and then I can leave you alone. I need your expertise, your charisma, your intelligence, your experience, your—“ His sister quickly cut him off.

“Oh shut it. I’m going to help you. However, only to get you out of my section as quickly as possible,” at his smile she finished rather bluntly, “so you don’t break anything.”

He nodded and promised to behave.“So what do you know about unit tangent vectors?”He looked from left to right hoping for inspiration to strike.“Erm…”“That’s what I thought. First the basics, you understand velocity correct?” He nodded. “Well,

velocity points in the direction of motion. This results in a smooth curve like an upside down parabola. If you have a smooth curve, you can take the slope. The slope of a line is the tangent of the line. So essentially if motion is projected as a smooth curve you’re guaranteed to have a unit tangent vector. Behind you, you’ll see the definition of a unit tangent vector,” she said pointing just behind him. He read it intently.

T(t)= r’(t) , r’(t)≠0||r’(t)||

“The big T is the unit tangent vector of t.”“Seems easy enough,” he said. His sister began to look very angry and he realized that he had

said the wrong thing.“Do you want my help or not?” She fumed.“No, no, I do. I’m sorry. I meant that I can understand that. What’s next?”She rolled her eyes at him.“Using unit tangent vectors you can derive the principal unit normal vector. A normal vector is a

perpendicular vector. Given a vector in the space, there are infinitely many perpendicular vectors. Our goal is to select a special vector, being the principal unit normal vector, which is normal to the unit tangent vector. This vector is the unique vector that points into the curve,” she pointed behind Vincent and he turned to see another definition below the first.

“There, now go find someone else to bother,” she said finished with him. Luckily for Vincent, she had explained everything in a way that he could understand. He would have to remember to come and patch things up with Ursula when he got out of the book and gained some worldly experience. His pressing dilemma at that moment, though, was who to ask his questions of next.

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T'(t) N(t) = ||T'(t)||

 

Chapter 7: Unit Tangent Vectors and Unit Normal Vectors“Father?” Vincent called out when he arrived back in his section. He had looked high and low

and could find nobody to teach him about arc length and curvature. His father suddenly appeared in front of him looking particularly haggard.

“Freshmen,” was his response to his son’s questioning look. “Well, Vincent. Did you find out all you need to know about being a vector-valued function?”

Vincent had his eyes downcast as he responded, “Almost. I still need to learn about the arc length of a space curve, the arc length function, definition of curvature, and formulas of curvature.”

“Oh, well that’ll be over very quickly son. I happen to be an expert on all of that! They don’t call me Arc length Arty for nothing!” Vincent looked at his father with an eyebrow cocked upwards.

“Dad, your names Curtis…?”“Arc length Arty, Curvature Curtis, all the same really. Sit down and I’ll explain it all and then

you’ll be ready to go.” Vincent quickly sat and waited the anticipation building. “Now arc length has to do with parametric equations, no matter what curve you’re on.

Whether, its plane or space, essentially it’s the same. The standard parametric equation says that x=x(t) and y=y(t), a≤ t≤ b is as follows:

“Since vector valued functions are parametrically defined curves in disguise, we have the same definition. Let r(t) = x(t) i + y(t) j + z(t) k be a smooth curve on an interval [a,b] then the arc length s is:”

At Vincent’s nod of understanding, his father continued.“Which brings us to the arc length function. Recall that like parametric equations, vector valued

functions can describe not only the path of a particle, but how it is moving. Through everything you’ve learned today there are many ways to represent this curve, but if we involve the definition of radians and say that the particle is traveling at a constant rate of one unit per second we can say the curve is parameterized by the arc length. On a unit circle, on e radian is equal to one unit of arc length around the circle. This gives us the arc length function. If r(t) is a differentiable vector valued function, then the arc length function is defined as follows:

“Where v is velocity, first derivative of r(t). How we doing so far?” His father asked, taking a moment to see if Vincent was absorbing everything. His son smiled and he continued.

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S

“Which leads us directly into curvature. Remember T(t) is the unit tangent vector function which defines the curvature at the rate at which the unit Tangent vector changes with respect to arc length. Curvature, essentially, describes how broad or narrow a curve is. If the curvature is zero then the curve looks like a line near this point. While if the curvature is a large number, then the curve has a sharp bend. This can be explained more readily by the definition of curvature.

||dT|| K(t) = ||ds|| = ||T'(s)||

“Keeping in mind that a circle has the same curvature at any point and that the radius of the circle and the curvature are inversely related. Which leads us into formulas of Curvature. If C is a smooth curve given by r(t) then the curvature of C at t is given by

||T’(t)|| K = ||r’(t)||

“And because r’(t)=ds/dt, the first formula suggests the ratio of the rate of change in the tangent vector T to the rate of change in arc length is curvature.”

Vincent nodded feeling at peace with himself. He had come along way and learned a great deal. “I think I’m ready, father,” he said with all the confidence in the world.“That’s good because I see that crazed professor flipping to our section. It’s your turn now son,

make me proud,” his father said as Vincent was being scrawled out onto the blackboard.In a book, on a shelf, in any given Calculus class, lives a very odd collection of functions. This

unique family provides the ability for motion to be represented along a curve or the graph of a curve to be traced. They are the family of Vector-Valued functions and they map real numbers to vectors. That is their purpose, while they serve many different roles in everyday life. Vincent the vector-valued function served his purpose in

The End.

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||r’(t)xr”(t)||= ||r’(t)||3