Differential calculus is a powerful mathematical tool
for analyzing how things change.
The handles of this tool
are a few simple rules for calculating derivatives.
Somewhere around 600 BC,
Someone discovered that if you want to make
pleasant-sounding chords on a string instrument
then the lengths of the strings ought to be a ratio of simple numbers, like 1:2, or 2:3, and so on.
These are called the 'Pythagorean harmonics'.
The discovery was very important because it was the very first discovery
of a connection between mathematics and the physical world.
Unfortunately, that connection was forgotten,
and had to be rediscovered slowly and painfully
thousands of years later.
One person would certainly grasp it though, was Galileo Galilei.
In that case, I want to read to you something that Galileo wrote.
This book was published in Rome, in 1623,
and is called "IL SAGGIATORE"
which is usually translated to mean "The Assayer."
But I preferred to call "The Experimentalist"
because I think that's more like what Galileo had in mind.
Galileo had this nasty habit
of making his famous remarks in Italian.
But I'll translate it for you as I go along.
He said: "True knowledge is written in this enormous book
which is continously open before our eyes.
I speak of the universe.
But one can't understand it unless first one learns to understand the language
and recognized the characters in which it is written.
It's written in a language of mathematics."
So, to prepare to read the book of the universe,
we must first learn its symbols and the vocabulary,
of the language of mathematics.
It's a language of precision
of poetry and even of music.
For a long time, physicists argued the language of mathematics.
And since somewhere around 600 BC,
so have musicians.
As with just about all languages, including music.
Mathematics has its own vocabulary,
its own rules and samples,
its precision and elegance,
its poetry and its history.
One part of that history was Galileo Galilei,
who was something of a nonconformist.
It was a trait he inherited from his father Vincenzo,
who was an accomplished musician.
Musically, Vincenzo refused to be bound by traditional forms.
A position that would become the family trademark.
He wrote a book composing theses of the Pythagorean Harmonics
among his musical contemporaries.
He considered the Ancient Greek chords too simple
for the complex musical structures of the Italian Renaissance.
Later, like father like son,
Galileo considered the Greek standards of mathematics too simple to express his ideas.
He created the science of kinematics,
a branch of mechanics dealing with motion and the abstract.
And if any abstract idea is to be expressed properly,
it needs appropriate language.
The concepts and symbols that give an idea its meaning and value
As advanced as it was,
Galileo's new science of motion
was still rooting in the soil of the Ancient Greek candlelight.
And for anything entirely new to bloom in the fields of mathematics and science,
scholars needed a language more sophisticated
than the one that had been spoken since Archimedes and Euclid.
In other words, after Galileo,
physics needed an advanced language.
Fewer than 25 years after his death,
that very language would be discovered,
and in one form or another, spoken ever after.
It would come to be called "Differential Calculus."
Differential calculus is very powerful.
And as with any language,
it derives its power from the idea behind it:
the derivative.
The derivative is to kinematics what the wheel is to travel.
A simple, yet very effective means
for getting from one place to another.
And, to get just the right perspective of what else the derivative is
Nothing works better than a little exercise.
To begin with, a derivative doesn't apply only to a body in horizontal motion,
nor for that matter, only to a body in vertical motion.
Up, down, any direction.
A derivative is the rate of change of any function,
at any exact point or estimate
As illustrated by Galileo's law of falling bodies,
Speed is the derivative of distance.
But it's more than that.
A derivative can represent the rate of change of anything.
For example,
The population density of dolphins
in relation to increasing or decreasing water temperature.
Or the rate of change in the volume of a balloon
versus its surface area.
Or even the rate of change in the price of a pizza with respect to its size.
Clearly then, the concept of the derivative goes far and wide.
But the mechanical process of the derivative,
differential calculus, needs a practical approach.
Or the concept goes nowhere fast.
Eventually, without the rules of differentiation,
the simple concept of the derivative becomes an uphill struggle.
In the long run, it helps to pick up a few more definitions along the way.
So before it's too late to turn back,
consider the factor of steepness.
On the incline,
the steepness is the ratio of the change in elevation,
the change in horizontal distance.
This ratio, a number, is called 'the slope'.
For example,
Suppose the elevation of an incline increases 15m every 100m
The rider moves upward 15, horizontally 100.
The slope is 0.15
A hill with a slope of 0.3
is twice as steep as one with the slope of 0.15
The bigger the slope, the steeper the hill.
When the slope is large, it's no small feat to get to the top.
When the slope is next to nothing, near 0, it's easy-going.
And when the slope is negative, it's downhill all the way.
Although mathematics can be easy-going,
it does have its peaks and valleys
And it always has.
Nobody knows who first asked the best way to get from here to there.
But the answer, in algebraic terms,
was first offered by a French mathematician named Fermat.
In 1629,
He came up with the idea of finding the tangent line to an arbitrary point on a curve.
In 1638,
Fermat shared his discovery with his friend and rival, René Descartes.
who had his own method for finding tangents to algebraic curves.
Many of these mathematical ideas,
especially those of Fermat
were further developed by Wilhelm Leibniz and Isaac Newton
into a general and systematic method of mathematical analysis:
Differential Calculus.
Setting history aside, at least for the present,
Some timely questions remain.
For example, on a smoothly changing curve,
there's a constantly changing slope.
How then, in today's language, is a slope calculated
at any given point?
To determine the slope at a particular point,
Here for example, simple take another point on the hill
it doesn't matter where.
Now connect the two points with a straight line.
That line is called a chord.
And its slope depends on the location of the second point.
If the first and second point are reasonably close,
The chord is a reasonably good approximation of the bike's path.
Now, move the second point closer to the first.
Move it even closer.
The slope is a number and as the points get closer together,
The number gets closer to a certain value.
It's reasonable to call that number 'the slope of the hill' at that point.
The line with that slope through the point is called 'the tangent line.'
And the tangent line is just what the chord turns into
as the points get closer together.
And the slope of the tangent line at that point
is the slope of the hill.
Instantaneous speed can be calculated along the same lines.
Galileo's law of falling bodies apply here to a rather relative one.
It's more than an experiment in detail.
It's differential calculus to the rescue.
Change in distance is divided by change in time.
The ratio is the average speed during a given time interval.
When that time shrinks to zero,
the limiting value of the average speed is the instantaneous speed.
Change in elevation is divided by change in horizontal distance.
The result is the slope of the chord joining 2 points.
When the horizontal distance shrinks to zero,
the limiting value of the slope of the chord is the slope at that point.
Differentiation could overcome calculations differ
but not the essential concept nor the process.
Speed is the derivative of distance with respect to time.
Slope is the derivative of elevation with respect to horizontal distance.
In any case, a derivative is what happens to a quotient.
A ratio of 2 numbers as both top and bottom shrinks to zero.
Before they actually reach zero,
small numbers are marked by the Greek letter Δ (delta)
Δy is the small change in y
Δx, a small change in x
So, Δy/Δx is merely a ratio of 2 small numbers.
When the small numbers shrinks to zero,
that ratio becomes a derivative.
And the deltas become a new symbol
dy/dx, the symbol of the derivative.
Which means, the derivative with respect to x of the quantity y.
Once this simple mechanics are mastered,
finding the derivative for just about anything
is no harder than flipping a switch.
The derivative of a function is a slope of its tangent at each point.
The derivative of a function is itself a function.
If the function is linear, the slope is constant.
And the derivative is just that constant.
If y=sin(x), then dy/dx=cos(x).
If y=cos(x), then dy/dx=-sin(x).
Taking derivatives takes a little practice,
but it's well worth the effort.
And, considering any number of contemporary derivative machines,
it has become a modern practice.
A speedometer is a derivative machine.
It measures the derivative of the distance traveled,
at each instant along the way.
The rate of change of position is the instantaneous speed
expressed at miles per hour (MPH).
Of course when the vehicle isn't moving,
no distance is being traveled.
Here, the position is constant.
And the derivative of a constant is zero.
Mathematics is a language built on grammatical structure
A collection of rules that both builds and breaks down
the composition of the task at hand,
whenever the masterwork.
From building a house to composing a symphony.
The most complicated task can be broken down in much the same way.
Newton and Leibniz developed the tools of calculus
that permit the most complex functions to be differentiated
by breaking it down into simple parts.
One of the basic rules of differentiation is the sum rule.
Suppose one painter can paint 90m² of wall per hour
And the other 100.
Those are the rates at which areas of the wall are changing color
In other words, they are derivatives.
Therefore, every hour, 190m² of wall are being painted altogether.
That's how the sum rule works.
The derivative of the sum
is the sum of the derivatives.
Another handy tool is the product rule.
which is used to find the derivative of the product of two functions.
For example, the area of any board
is the product of its length times its width.
If the length is shortened,
the change in area is the width times the change in length.
If the width is reduced,
the change in area is the length times the change in width.
The total change in area is the sum of these.
That's true of the carpenter's product
and is just as true in the language of differential calculus.
The derivative of the product yz
is y times the derivative of z
plus z times the derivative of y.
Using this rule, it's possible to find the derivative of x²
Or of x³
Or of any power of x
The derivative of xⁿ is n.x^(n-1)
Often when an operation is depended on another.
For example, suppose the vehicle has a fuel efficiency of 17 miles per gallon.
That, too, is a derivative.
If y is the distance traveled, and x the amount of fuel consumed
Then, 17 miles per gallon equals dy/dx.
Say it uses 2 gallons every hour,
2 gallons per hour equals dx/dt.
A vehicle's speed in miles per hour is equal to
the miles per gallon it gets times the gallons per hour it uses.
This is the chain rule.
It's used when y depends on x
and x depends on t.
The sum rule
The product rule
And the chain rule
These 3 rules represent the grammar of differential calculus.
And the value of differential calculus
can be seen in the variety of its applications.
For example, when a rocket moves with displacement s and time t
The derivative of the displacement is the velocity
positive for upward motion
and negative for downward motion.
The derivative of the velocity is the acceleration
which is the same as taking the derivative of a derivative.
That is, the second derivative of s.
The acceleration is caused by the firing of the rocket.
The rules of differential calculus
and their applications to physics
This stands as the solo instrument
and play in the art and science of mathematics.
Working together, harmonizing, they can blend invidual notes or numbers
into an exquisite harmony.
I just got a letter from a musician named Albert Einstein.
He sent it in 1912.
The mail was lost some times.
But that's not the reason that I just got it,
the reason is because he didn't send it to me, he send it to a friend of his
named Arnold Sommerfeld. And I just got it in the library.
Anyway, I wanted to read to you a little bit of what he wrote.
He said: "I occupy myself exclusively with the problem of gravitation.
And now, believe I'll overcome all difficulties.
Because one thing is certain
I've become imbued with a great respect for mathematics
The subtle parts of which, in my ignorance,
I had, until now, regarded as pure luxury."
Einstein worked on gravitation for 4 more years
And what came out is called "the general theory of relativity"
It is the most fiercely difficult mathematical theory in all of physics.
So what did Einstein mean by saying that
the subtle parts of mathematics had seemed a luxury?
Did he really believe that he was going to get along without doing any calculations?
Well of course not.
The point is that, physicists have a certain arrogance about mathematics.
For example, in today's lecture,
You must have gotten the impression that all you have to do
is follow some simple rules
and you can take the derivative of any function in the world.
Well that's not quite true.
Suppose you had a function
which looks like an Egyptian pyramid.
Well, it's easy to see what the slope is here.
And it's easy to see what the slope is here.
But right here at the peak, you'd be in trouble.
Because it has no slope at that point.
The function has no derivative at that point.
Now I never told you anything that would lead you to believe
that that could ever happen.
You see, for physicists, mathematics is just the tool
It's to be used in order to accomplish something else.
But a real mathematician
is the guardian of precision and clarity of thought
What interests the mathematician is the mathematics itself.
When a mathematician makes a statement about derivatives,
the statement takes into account every exception
no matter how bizarre or unusual, like the peak of the pyramid.
That's the kind of subtlety
that Einstein was worried about.
I'll see you next time.
Subtitle created by Tran Nguyen Phuong Thanh - 2013.