Tip:
Highlight text to annotate it
X
>> MANY THINGS IN THE UNIVERSE
BEHAVE IN A SYNCHRONIZED WAY --
WHETHER MAN-MADE OR NATURAL --
WORKING TOGETHER IN HARMONY,
MOVING SIMULTANEOUSLY, PULSING
WITH A REGULAR RHYTHM,
COORDINATED IN TIME AND SPACE.
"MOVING IN SYNC," WE WOULD SAY.
WE SEE SYNCHRONIZATION AS AN
EMERGENCE OF SPONTANEOUS ORDER
IN SYSTEMS THAT MOST NATURALLY
SHOULD BE DISORGANIZED.
AND WHEN IT EMERGES, THERE'S A
BEAUTY AND A MYSTERY TO IT,
QUALITIES THAT OFTEN CAN BE
UNDERSTOOD THROUGH THE POWER OF
MATHEMATICS.
HOW DOES A SYMPHONY ORCHESTRA
PLAY IN SYNC AND HARMONY?
THE ANSWER IS OBVIOUS: THE GROUP
CONSISTS OF HUMANS WHO HAVE THE
CAPACITY TO READ MUSIC, LISTEN
TO EACH OTHER, AND FOLLOW THEIR
LEADER.
BUT WHAT ABOUT A GROUP WITH NO
LEADER OR SHEET MUSIC?
LIKE A JAZZ JAM SESSION?
ITS MEMBERS STILL PERFORM IN
SYNC, EACH PERFORMER WORKING OFF
THE CUES OF THE OTHERS,
SYNCHRONIZED VIA SPOKEN OR
UNSPOKEN COMMUNICATION.
WHAT ABOUT SCHOOLS OF FISH OR
FLOCKS OF BIRDS?
HOW DO THEY KNOW WHEN TO TURN
LEFT, TURN RIGHT, MOVE HIGHER OR
LOWER, PERFECTLY IN SYNC, EACH
ADJUSTING ITS MOVEMENTS
INSTINCTIVELY, SOMEHOW HIGHLY
SENSITIVE TO WHAT ITS NEIGHBORS
ARE DOING?
WHILE THE ORCHESTRA AND MARCHING
BAND WORK IN A PREMEDITATED,
PLANNED, EVEN CALCULATED ORDER,
THE FLOCK OF BIRDS AND SCHOOL OF
FISH, THEY MOVE IN UNISON BY A
PHENOMENON CALLED "SPONTANEOUS
BIOLOGICAL ORDER."
IN THEIR COLLECTIVE MOVEMENT, WE
SEE A SYSTEM OF INDIVIDUALS WHO
SOMEHOW HAVE MANAGED TO
SYNCHRONIZE THEIR CHANGES IN
MOTION.
AND THROUGH MATHEMATICS, WE CAN
MAKE SENSE OF THIS GROUP DYNAMIC
USING THE LANGUAGE OF CALCULUS.
TO PUT IT SIMPLY, CALCULUS
ALLOWS US TO MAKE MATHEMATICAL
SENSE OF CHANGE IN MOVING
SYSTEMS, WHETHER GRADUAL OR
CONSTANT.
FOR INSTANCE, A CAR'S
SPEEDOMETER MAY SAY 55 MILES PER
HOUR, BUT THAT'S JUST A ROUGH
APPROXIMATION OF ITS EXACT SPEED
AT THAT MOMENT.
HOW CAN WE DESCRIBE EXACTLY HOW
FAST THE CAR IS MOVING AT EACH
INSTANT IT DRIVES ALONG?
FOR THIS ANSWER, WE MUST GO BACK
TO THE 17th CENTURY, WHEN THE
ENGLISH SCIENTIST ROBERT HOOKE
CHALLENGED HIS RIVAL ISAAC
NEWTON WITH THAT KIND OF
QUESTION.
HE ASKED: HOW EXACTLY DO THE
PLANETS PULL ON EACH OTHER, AND
DOES THE LAW OF PULLING EXPLAIN
THE ORBITS THAT WE SEE?
TO ANSWER IT, NEWTON REALIZED
THAT HE HAD TO DESCRIBE AND
MEASURE THE MOVEMENT OF THE
PLANETS INSTANT BY INSTANT.
AND SO, IN 1666, HE
INDEPENDENTLY CREATES A NEW
BRANCH OF MATHEMATICS, CALCULUS.
NEWTON NEEDED A WAY TO NOTATE
THE INCREMENTAL CHANGES OF WHERE
A PLANET WAS ON ITS ORBIT AND
THEN TO BE ABLE TO ADD UP ALL OF
THOSE INCREMENTAL CHANGES TO
ACTUALLY CONSTRUCT THE ORBIT.
HE ALSO NEEDED A WAY TO FIGURE
OUT HOW FAST IT WAS MOVING AND
WHERE IT WOULD MOVE NEXT.
HE SAW UNITY; GRAVITY DIDN'T END
AT THE ATMOSPHERE.
NEWTON WAS THE FIRST TO SHOW
THAT THE MOTION OF OBJECTS ON
EARTH AND PLANETARY MOTION ARE
GOVERNED BY THE SAME SET OF
NATURAL LAWS.
HIS FAMOUS EQUATION "FORCE
EQUALS MASS TIMES ACCELERATION,"
F=MA, NEATLY SUMS THAT UP.
THIS SAME MATH HE USED TO STUDY
THE MOVEMENT OF INANIMATE
OBJECTS IN THE 17th CENTURY
TODAY ASSISTS US IN CALCULATING
THE BEHAVIOR OF ANIMATE OBJECTS.
THANKS TO NEWTON, CALCULUS
PROVIDES A MATHEMATICAL
LANGUAGE THAT ALLOWS US TO
MEASURE A VARIABLE.
FOR INSTANCE, FOR A BIRD, THE
ANGLE OF FLIGHT, OR FOR A CAR,
ITS SPEED.
BUT BIRDS AND DRIVERS ARE THEIR
OWN ENTITIES WITH REACTIONS AND
INFLUENCE FACTORS, AND THEREFORE
REPRESENT MOVING OBJECTS WITH
MANY VARIABLES.
IT'S PERHAPS A BIT EASIER TO
LEARN ABOUT CALCULUS BY
EXAMINING THE SOLAR SYSTEM,
BECAUSE WITH PLANETS THERE'S A
CONSISTENT MEASURABLE FORCE AT
PLAY.
IN ORDER TO STUDY THE RATE OF
CHANGE OF AN OBJECT IN MOTION,
LIKE A PLANET, WE LOOK AT THE
TINY CHANGES IN THE POSITION OF
THE PLANET IN THE SKY.
THESE TINY CHANGES ARE THEN
CALLED "INFINITESIMALS."
THEY ARE A THEORETICAL CONSTRUCT
AND ARE MEANT TO BE THE SMALLEST
POSSIBLE INCREMENT BY WHICH A
QUANTITY CAN CHANGE.
THIS IDEA LEADS US ON TO
DERIVATIVES, ANOTHER IMPORTANT
CONCEPT IN DIFFERENTIAL