This is why you're learning differential equations
Summary
TLDRThis video explores the importance of differential equations in understanding natural phenomena and real-world applications. It delves into how these equations model population growth, fluid dynamics, electromagnetism, and even the motion of orbiting bodies. The script uses examples from TV shows, gym exercises with chains, and the spread of the coronavirus to illustrate how differential equations describe complex systems. The video encourages viewers to learn more about these foundational mathematical tools through Brilliant's courses, which focus on practical applications.
Takeaways
- đ We learn differential equations because they describe how nature and the universe work, often appearing in complex systems like population growth, fluid dynamics, and electromagnetism.
- đ Differential equations can be challenging to solve, but they provide valuable insights into various phenomena, such as modeling population growth or the movement of fluids.
- đ Maxwell's equations, which are four fundamental differential equations, are crucial for understanding electromagnetism, the basis for technologies like phones, radio, Wi-Fi, and GPS.
- đ Differential equations can be interpreted to find specific solutions, such as the unique curve that has an area under it twice the numerical value of its arc length over any interval.
- đ The script uses the TV show 'Numbers' to illustrate how differential equations can model real-world scenarios, such as the pursuit curve used by the FBI to predict criminal movements.
- đ Pursuit curves are used in various contexts, including missile guidance systems and aircraft navigation, and are derived from differential equations considering the motion and direction of chasing and chased objects.
- đïžââïž The video discusses the complex dynamics of exercising with chains, which involves a second-order nonlinear differential equation due to the changing mass as the chain is lifted.
- đŠ The spread of infectious diseases like COVID-19 can be modeled using differential equations, which consider the rates of change in susceptible, infected, and recovered populations.
- đ± Population growth models, such as the SI model for infections, are simplified differential equations that help predict how diseases spread and can be controlled through various interventions.
- đ€ The script highlights the importance of understanding the stories and meanings behind differential equations, which can be applied to a wide range of fields, from physics to biology and epidemiology.
- đ Brilliant.org offers courses on differential equations that focus on real-world applications, providing a comprehensive learning experience for students and enthusiasts alike.
Q & A
What is the primary reason for learning differential equations according to the video?
-Differential equations describe how nature works and are fundamental in modeling various real-world phenomena, such as population growth, fluid movement, and electromagnetism.
Can differential equations always be solved exactly?
-No, differential equations are often tough and sometimes impossible to solve exactly, but they can still provide valuable insights into the phenomena they model.
What are Maxwell's equations, and why are they important?
-Maxwell's equations are a set of four differential equations that describe electromagnetism, which is fundamental to how phones, radio, Wi-Fi, and GPS work.
How do differential equations relate to the motion of objects in physical contact or in orbit?
-Differential equations describe the motion of objects when a force is exerted, whether through physical contact or gravitational force, as seen in orbiting bodies.
What is a pursuit curve, and how is it used in real-world scenarios?
-A pursuit curve is the path traced by one object chasing another. It can be applied in scenarios such as a cheetah chasing a gazelle or in missile guidance systems, aircraft, and submarines.
How do chains in gym equipment illustrate the complexity of differential equations?
-Chains in gym equipment add weight as the bar moves up, creating a scenario where the mass changes with height. This requires a second-order nonlinear differential equation to describe the motion.
What does the SI model represent in epidemiology?
-The SI (Susceptible-Infected) model represents the transition of individuals from susceptible to infected in a population, helping to predict the spread of infectious diseases.
How can differential equations model the spread of a virus like COVID-19?
-Differential equations can model the spread of a virus by describing the rate of change of susceptible, infected, and recovered individuals in a population over time, as seen in the SIR model.
What is the significance of population growth models in differential equations?
-Population growth models use differential equations to describe how a population grows at a rate proportional to its current value, incorporating factors like birth, death, and interactions between species.
How does the video illustrate the real-world application of differential equations in engineering and physics?
-The video discusses various real-world applications, including modeling the motion of rockets with changing mass, the spread of diseases, and the behavior of systems like beams and waves, showing the practical importance of differential equations in engineering and physics.
Outlines
đ Introduction to Differential Equations
This paragraph introduces the importance of differential equations in understanding the workings of nature and the universe. It highlights their applications in various fields such as population growth modeling, fluid dynamics, electromagnetism (Maxwell's equations), and forces in orbiting bodies. The paragraph also touches on the narrative aspect of equations, using an example that relates to the area under a curve and its arc length, leading to the derivation of a specific differential equation. The discussion emphasizes the real-world significance of these mathematical tools, setting the stage for further exploration in the video.
đ Pursuit Curves in Real-World Scenarios
This section delves into the concept of pursuit curves, using a TV show scenario where the FBI tracks criminals' movements to predict their next actions. It explains how differential equations can model the path of a pursuing object, assuming a known path for the target and the pursuer's constant direction towards the target. The paragraph outlines the mathematical process of deriving a pursuit curve, involving vector representation, normalization, and the use of dot products. It also discusses variations of pursuit curves and their applications in missile guidance systems, aircraft, and submarines, showcasing the practicality of differential equations in various contexts.
đïžââïž The Physics of Chain-Loaded Exercises
This paragraph explores the complex physics involved in exercises using chains, such as bench presses or squats, where the weight increases as the barbell is lifted. It explains how the changing mass of the lifted chain segment affects the equation of motion, leading to a second-order nonlinear differential equation. The discussion simplifies the scenario by assuming a barbell with no mass and a constant lifting force, highlighting the need to consider changing mass in the analysis. The paragraph emphasizes the broader implications of analyzing systems with variable mass, such as rocket propulsion, where mass changes due to fuel consumption.
đĄ Modeling the Spread of Infectious Diseases
This section discusses the application of differential equations in modeling the spread of infectious diseases, using the coronavirus pandemic as a case study. It introduces the basic SIR (Susceptible, Infected, Recovered) model, which describes the dynamics of disease spread through a population. The paragraph explains how the rates of change for each category are determined by the interactions between them and external factors like social distancing and recovery rates. It also touches on the use of phase portraits and simulations to understand disease dynamics without solving the equations explicitly, providing a foundation for further exploration of population dynamics and disease modeling.
Mindmap
Keywords
đĄDifferential Equations
đĄMaxwell's Equations
đĄPursuit Curves
đĄSI Model
đĄEpidemiology
đĄVector
đĄDot Product
đĄNormalization
đĄSecond-Order Nonlinear Differential Equation
đĄPhase Portrait
đĄBrilliant
Highlights
Differential equations are fundamental in describing how nature and the universe work.
They are often challenging to solve but provide valuable insights into various phenomena.
Differential equations are used to model population growth, fluid dynamics, and electromagnetism, including the principles behind phones, radio, Wi-Fi, and GPS.
Maxwell's equations, consisting of four differential equations, are key to understanding electromagnetism.
Differential equations can describe motion with or without physical contact, such as the orbits of celestial bodies.
They can be given meaning by interpreting the story they tell, like relating to the area under a curve and its arc length.
An example of a pursuit curve from the TV show 'Numbers' demonstrates how differential equations can be applied to real-world scenarios like law enforcement tracking criminals.
Pursuit curves illustrate the path of one object chasing another and can be mathematically modeled using differential equations.
The video explains the mathematical process of determining the curve traced by a pursuing object, involving assumptions about the chase dynamics.
Differential equations are used to model the changing weight experienced during an exercise with chains, such as bench press or squat.
The motion of lifting a chain is represented by a second-order nonlinear differential equation, highlighting the complexity of seemingly simple movements.
The concept of changing mass in a system, like a rocket losing mass as it exhausts, leads to differential equations that describe its motion.
The spread of the coronavirus is modeled using differential equations, illustrating how the infection rate changes over time.
The SIR model is introduced as a basic framework to understand the dynamics of disease spread in a population.
Differential equations are essential in modeling population growth, including complex interactions between different species or factors.
Phase portraits are mentioned as a tool for visualizing the behavior of systems described by differential equations without needing an explicit solution.
Brilliant.org offers courses on differential equations that focus on real-world applications, including pursuit curves, wave equations, and beam behavior.
The video concludes with an invitation to learn more about differential equations and their practical uses through Brilliant's courses, offering a discount for the first 200 subscribers.
Transcripts
this video is sponsored by brilliant
let's just get to the point why do we
learn differential equations because
they are the equations that describe how
nature works the universe one day said
boom here's a bunch of stuff and we said
cool how does it all work and the
universe said differential equations
they're tough often impossible to solve
exactly but you'll eventually get some
cool stuff from them modeling how a
population will grow involves
differential equations how any fluid
moves differential equations
electromagnetism which is how phones
radio Wi-Fi and GPS all work not one but
four differential equations known as
Maxwell's equations there's electric
circuits and even if something touches
something else and thus exerts a force
differential equations are used to
describe the motion if there's no
physical contact like with orbiting
bodies there's still a force so there's
still differential equations now as with
simple algebraic equations differential
equations can and often do have meaning
when you read into the story that
they're telling like this equation could
be saying my age is y and I'm exactly
five years older than my brother whose
age is X and we can use this to
determine either of our ages given the
other pretty boring but we can give it
meaning this equation can have meaning
too or rather it asked the question and
that question is what curve or family of
curves have the property where the
numerical value of the area under the
curve is twice numerical value of the
arc length on that same interval for any
given interval A to B
well to set this up we start with this
equation which is the area under the
curve itself and this is the equation
for the arc length on that same interval
A to B but we want to know when the left
side always equals twice the right side
now we can differentiate both sides to
remove the integral sign and we're left
with a differential equation from here
we can square both sides and then
distribute and we get that original
equation if you solve for that function
y you get this which is the curve you're
seeing here so that's one example of
meaning within a differential equation
but let's see how these really describe
some real-world situations because it's
not always obvious what story these
equations tell or how they show up in
general and since I always love bringing
up the tv-show numbers when I can here's
a perfect example so check out this
scene from an episode where law
enforcement is trying to catch a couple
committing crimes as they travel across
the United States we plot your movements
against those targets the pattern makes
itself known and when we plot your path
against oil and winters we got this yeah
this is the Red Desert Robbery the
missing point on a curve that I didn't
even realize I was looking at it's a
variation on something called a pursuit
curve alright so what's happening here
is the FBI has been chasing this couple
across the country but hasn't caught
them yet so the mathematicians plotted
the path both the criminals and the FBI
took to see if they can make some
predictions about where the criminals
will strike next and this is related to
pursuit curves a pursuit curve is simply
the curve traced out by one object
chasing another although there are
usually conditions listed for something
to be technically considered a pursuit
curve
but this could apply to a cheetah
chasing down a gazelle or one aircraft
chasing another for example so let's see
how we can determine the curve that the
pursuing object will trace act now two
things to note first we're going to
assume that the plane being chased has a
predetermined path whether it's flying
straight up or in a circle or whatever
assume we already know their path then
the second assumption like I mentioned
before is that the chasing object is
always moving in the direction of the
other and it turns its nose as needed
during pursuit in reality this could be
like a plane wanting it's forward-facing
guns always aimed at the target or
something anyways what you're seeing
here is a snapshot of the chase and we
can represent the current positions of
each plane with a vector I'll say C and
M for cat and mouse the other thing we
know is that at this time or really any
time the chasing plane is pointed at the
front one which means this is their
velocity vector or C prime at this
moment there's no wind or anything so
the plane is definitely flying in the
direction it's pointed now we don't know
the speed or the length of that vector
currently but we know the direction at
least that's more important because
there's another vector pointing in that
same direction which is easy to find and
that's M minus C for the visualization M
minus C is the same as M plus negative C
and putting negative C on top of M gives
us a vector that points from the back
plane to the front one
then to deal with the lengths I'm gonna
normalize the two vectors by dividing by
their magnitudes so they both have a
length of one this is key because the
dot product of two vectors both with
length one and pointing in the same
direction is one this is a fundamental
equation yes I could have just set the
vectors equal to one another but I'm
doing this because in that episode when
the mathematician is explaining pursuit
curves you can see that equation come on
the screen so now you know the meaning
behind it but still it's not really
obvious how to solve it yet but if we go
back to our snapshot remember that the N
vector is actually a known function of
time we're seeing it only at one moment
in time but it is changing as the planes
fly around so I'll say it's a vector
function U of T comma V of T which are
both known I'm just keeping things
generic the C vector also changes in
time but it's our unknown some X of T
comma Y of T that we want to solve for
and the C prime vector would just be X
prime of T comma Y prime of T so then M
minus C would give us this vector here
just the X components subtracted and the
Y components subtracted and I'm just not
writing the t's so there's room I'll do
the same thing with C prime then if we
plug all of those into our equation that
we want to solve we get this now the one
extra thing I did was set the magnitude
of C prime to 1 which just means we
assumed the chasing plane is flying at a
constant speed of 1 just to simplify the
equation then we just have to do the dot
product which leaves us with this
differential equation
you'll notice I actually wrote out the
expression for the magnitude of M minus
C on the bottom here the only problem
with this is that there are two unknown
functions x and y which means we need
another equation but that would just be
the one saying the chasing plane is
flying at a constant speed of 1 now we
have a system of differential equations
that can be solved if for example we
assumed the target plane is flying
straight along the y axis then this
would be the path of the chasing plane
shown in red if the target plane we're
flying in a circle then you get
something very different shown here on
brilliant sight now you'll notice this
method of chasing someone isn't
necessarily ideal for catching them but
there seems to be other types or
variations of pursuit curves that range
from always aiming at the target to
predicting where they'll go next on that
numbers episode the situation was more
complex as the mathematicians were
accounting for how the Meuse of the FBI
might affect the criminals that were
being chased but still the basics of
pursuit curves can be seen in a first
level differential equations course and
while this might not be a real-world FBI
case pursuit curves can be applied to
missile guidance systems aircraft
submarines and so on all right now let's
look at something more casual if you go
to the gym you may have seen or done a
bench press or squat with chains hanging
off the sides this makes it so that as
the bar moves up the chain is more and
more suspended and this contributes more
of its weight to the exercise meaning as
the person pushes upwards the weight
increases this actually complicates the
equation of motion more than you might
think
because for every little DX or change in
height yes I'll be using X as the
variable there's a DM or small change in
mass in regards to the part of the chain
that's off the ground so let's see what
this set up would look like
now we'll say the barbell has no mass
just to make things easier so really
we're just lifting the chain off the
ground and we'll call that distance off
the ground X measured in meters then
let's give the chain a way to density of
10 Newton's per meter
thus the weight of just the section off
the ground is 10 X so if the chain were
2 meters off the ground then you'd have
to use a force of 20 Newtons to hold it
in place then the equation for mass for
the part of the chain off the ground is
simply the weight divided by gravity
weight is 10x and I'll round gravity to
10 as always so the mass of this part of
the chain off the ground is just X
lastly we'll say the person is pushing
up with a constant force of 50 Newtons
meaning the net force is 50 up minus the
10x down from the chain itself okay now
before we can move on we have to realize
that F equals MA is a lie well not
really but it only applies to special
cases where the mass is constant the
real equation we have to work with is f
equals the rate of change of momentum or
mass times velocity doing a simple
product rule we get this here and you'll
notice when M is not changing which we
get used to in a first level physics
course then this term is zero and we're
left with F equals m times dv/dt or F
equals MA as usual but with the changing
mass we have this entire equation from
here I'm just going to replace the
variables like mass is really X and
force is 50 minus 10x so we get this
here but velocity is just the rate of
change of position and DV DT which is
acceleration is the second derivative of
position so we're left with this
equation here moving the 10x over we're
left with a second order nonlinear
differential equation solving this would
not be easy but not really the point of
this video instead I just want to
highlight that the motion which results
by something as simple
pulling a chain upwards has to be
expressed through a not so simple second
order differential equation and while it
may be true that analyzing the motion of
a barbell isn't too applicable the idea
of analyzing a system with changing mass
is the perfect example is a rocket as
exhaust leaves the bottom the rocket
itself loses mass little by little so
thrust and a changing mass are kind of
linked and the situation also leads to
differential equations but now let's
look at the most real-world application
I can think of this here is a curve of
the number of currently infected people
from the corona virus as of early June
it's infected people versus time which
means the instantaneous rate of change
or slope at any point is di DT it
gradually increased for a while before
flattening out but we want that rate of
change to go negative anyways this is an
incomplete picture of what's going on
because there are other categories of
people out there in the population there
those that have never been infected or
those that are susceptible those that
are currently infected and then those
that are recovered or unfortunately
deceased but the most basic model the SI
R model just calls it recovered and
assumes no deaths I know many youtubers
have talked about this recently so I'll
keep it kind of short in the case of the
corona virus everyone started in the
susceptible bucket and let's say the
population is 20 so no one is infected
but everyone has the potential to be and
then one day one person transition to
infected this was basically an initial
condition now if the population stays
constant no one new is born then the
number of susceptible people can only go
down or stay the same because as soon as
you get infected you leave that group
never to return
assuming immunity after you get the
disease so the rate of change of
susceptible people is going to be
negative something it can only go down
since having a lot of infected people or
a lot of susceptible people / a high
population
increases the magnitude of that change
then we say it's s times I where s and I
are the number of susceptible and
infected people respectively if either
of these are large and the other is
nonzero you have a large transition to
those who are infected but we also have
to include a constant that constant
depends on the virus and us as it scales
how quickly people go from healthy to
sick social distancing or hand-washing
for example would decrease that constant
and this rate of change wouldn't be as
extreme then the rate of change of those
who are infected starts with that same
expression seen on the left but positive
this is because in this model it's a
one-way street if someone leaves the
susceptible pile they go to infect it
but people will leave the infected pile
proportional to how many infected people
there are this constant in reality is
death rate combined with recovery rate
if medicine is released that speeds up
recovery by 50% and that constant goes
up and people are quicker to leave the
infected pile and become recovered the
rate of change of those who have
recovered can only be positive or zero
and it's that same constant times I
you'll notice in this model that we kept
the population constant so the sum of
the three categories was always the same
meaning the rates of change should add
to zero as they do but here we're left
with another system of differential
equations that when solved will tell you
how an infection will spread through a
population in this simplified scenario
I'm sure many of you have seen number
files video on this that I'll link below
but there you can see what happens when
you play around with the equations and
constants and all that as with most of
these examples we did simplify things to
make the math easier but this is all the
foundations of what's going on in the
real world if you go to the website for
the International Council for industrial
and applied mathematics there's a page
on the mathematics of koban 19 showing
the simulations and models that
mathematicians are creating to
understand the spread of this virus in
different parts of the world where
you'll find equation
just like we saw equations very similar
to these also show up in terms of
population growth one of the first
differential equations a lot of us learn
is the one that models how a population
will grow at a rate proportional to its
current value the more people or animals
there are the faster the growth as
expected but things get more complicated
with deaths or when maybe one population
kills off another here's an example of
bacteria which can multiply on their own
and phages
that essentially feed off bacteria and
thus will die without them and you'll
see that after one cycle here the phages
grow from 1 to 4 but the bacteria stay
at 2 and now there are more phages so
some will die while others will continue
to multiply so there's kind of this
back-and-forth that happens but it leads
to differential equations where the
rates have changed depend on multiples
of the populations and it's not always
about just solving the equations as
there are tools such as phase portraits
that can help paint a picture about
what's going on without having to find
an actual solution here if you're given
a certain population of phages and
bacteria this would tell you how the
system or really both populations will
change at that moment and with this we
can find equilibrium points or long term
behavior for example and while I'm not
gonna go any further than this think
we've discussed a lot if you want to
dive more into the topic of differential
equations you can of course do so right
here at brilliant currently they have
two differential equations courses which
are some my favorites because of how
much they focus on real-world
applications they have the pursuit
curves we discussed there's 2d and 3d
wave equations
there's the equations that model the
behavior of beams and much more their
first course does start at the basics
for anyone just starting out but by the
second course there are things I never
saw as an engineer in college so
regardless of where you are in your
education there likely is a lot to learn
whether you want to get ahead as a
student or just brush up on old topics
so if you want to get started right now
support the channel you can click the
link below or go to brilliant org slash
Zack star plus the first 200 people to
sign up will get 20% off their annual
premium subscription and with that I'm
going to end that video there thanks as
always my supporters on patreon social
media links to follow me are down below
and I'll see you guys in the next video
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