Skip to main content
Loading Explore guides…
Differential Equations · Acedra
Home
Explore
Mathematics
Differential Equations
Sign in to save
Chat
If YouTube asks you to sign in, open the video on YouTube.
Watch on YouTube
Reload player
Transcript
287 segments
Copy
Hide video
0:01
Pierre Albin: Hello and welcome to differential equations.
0:07
Pierre Albin: In this video, we're going to talk about what differential equation is and different classifications that will be useful, this semester
0:16
Pierre Albin: for the class syllabus and the logistics of how the class will be run. You can consult the course, web page and the first day of classes.
0:28
Pierre Albin: So what we want to talk about now is what
0:33
Pierre Albin: is a differential equation.
0:44
Pierre Albin: Well, the differential equation is an equation
0:54
Pierre Albin: where the unknown.
1:01
Pierre Albin: it's a function.
1:09
Pierre Albin: And we're trying to figure out.
1:19
Pierre Albin: And what we are given
1:27
Pierre Albin: is some relationship
1:39
Pierre Albin: between the function and its derivatives.
1:55
Pierre Albin: We call a differential equation.
2:07
Pierre Albin: an ordinary
2:18
Pierre Albin: differential equation
2:21
Pierre Albin: abbreviated ode.
2:27
Pierre Albin: if the derivatives
2:36
Pierre Albin: are taken with respect to only one variable.
2:55
Pierre Albin: So, for example.
3:03
Pierre Albin: so really simple differential equation. Pierre Albin: we just know that the derivative of y with respect to t is 0. Pierre Albin: Well. Pierre Albin: then, we know that y is a constant
3:17
Pierre Albin: right.
3:18
Pierre Albin: almost as simple. If what we know is that the third derivative of y with respect to t is equal to 0, Pierre Albin: then we know that y is a polynomial of degree at most 2,
3:35
Pierre Albin: so we can write y as some constant c, 0 plus some constant C one times t plus some constant C 2 times T squared. Pierre Albin: Right? Of course, most differential equations are not anywhere near this simple, we might have something like Pierre Albin: why plus
3:54
Pierre Albin: take the first derivative of y with respect to t square it. and then take the eleventh derivative Pierre Albin: of Y with respect to T, and maybe that's equal to cosine T, Pierre Albin: so equations like this, we have no hope of being able to write down an explicit solution.
4:14
Pierre Albin: But in each of these cases the unknown is some function y. which is a function of T.
4:23
Pierre Albin: These are all ordinary differential equations, because we are only taking derivatives. With respect to T,
4:37
Pierre Albin: we call a differential equation.
4:48
Partial differential equation
4:59
Pierre Albin: abbreviated. Pd.
5:08
Pierre Albin: Gifted. The derivatives
5:12
Pierre Albin: are taken
5:19
Pierre Albin: with respect to more than one.
5:27
Pierre Albin: Very well.
5:30
Pierre Albin: So, for example.
5:38
Pierre Albin: let's say you have some function. U, and you know that is, derivative respect to T
5:44
Pierre Albin: is equal to
5:47
Pierre Albin: second derivative of a view with respect to X plus the second derivative view. With respect to Y, this, an example of a heat equation.
5:58
Pierre Albin: we'll talk about partial differential equations near the end of the semester. We're going to spend most of the semester talking about ordinary differential equations.
6:08
Pierre Albin: Okay? So we have that division into ordinary and partial differential equations.
6:14
Pierre Albin: We further classify differential equations by their order.
6:44
Pierre Albin: The order
6:49
Pierre Albin: of a differential equation
7:01
Pierre Albin: is the order.
7:09
Pierre Albin: the highest derivative.
7:22
Pierre Albin: which appears in the equation.
7:27
Pierre Albin: So, for example.
7:31
Pierre Albin: if you have an equation that looks like DYDT,
7:37
Pierre Albin: the fifth power Pierre Albin: plus Pierre Albin: sine of the derivative with y, with respect t
7:46
Pierre Albin: plus second derivative of y. With respect to t. Pierre Albin: it's equal to E to the t.
7:55
Pierre Albin: then this is a second order, differential equation. Pierre Albin: because even though here you're taking the fifth power. And here you're taking sign of this function.
8:06
Pierre Albin: This you're only using the first derivative of Y here, you're only using the first derivative respect to y, so this far, the equation would only be first order here. You're taking 2 derivatives of Y, and so the equation Pierre Albin: is second order.
8:27
Pierre Albin: If we scroll back to these equations, then of course, this equation is a first order, differential equation. Pierre Albin: This equation is a third order, differential equation. Pierre Albin: This equation is an eleventh order. Differential equation. Notice over here, first order, differential equation, and we ended up with one constant
8:50
Pierre Albin: third order, differential equation, and we ended up with 3 constants.
8:57
Pierre Albin: That is not a coincidence.
9:00
Pierre Albin: The order is generally equal
9:14
Pierre Albin: to the number
9:19
Pierre Albin: of constants that you have to specify in order to figure out a solution Pierre Albin: or in order to specify a solution uniquely. Another way of thinking about that is that it's equal to the number of initial conditions
9:44
Pierre Albin: that are required
9:49
Pierre Albin: to specify.
9:57
Pierre Albin: For example, if we had.
10:08
Pierre Albin: let's say we had that same differential equation.
10:12
Pierre Albin: just knowing the third derivative is equal to 0. And let's say that we're also given that y of 0 is equal to 17 derivative of y with respect to t Pierre Albin: t equal to 0 is equal to 5, and the second derivative of y with respect to t at T equal to 0 is equal to minus 4, then we would know
10:38
Pierre Albin: that Y is not just some arbitrary polynomial of degree. 2. It is Pierre Albin: this
10:46
Pierre Albin: polynomial of degree. 2. Right? So as soon as you give me 3 initial conditions, then I can specify those 3 constants that we had in the general solution Pierre Albin: right and figure out a unique solution.
11:05
Pierre Albin: a differential equation such as this one
11:20
Pierre Albin: where all of the data
11:27
Pierre Albin: is specified
11:33
Pierre Albin: at 1 point
11:38
Pierre Albin: is an initial value problem.
11:51
Pierre Albin: If the data
12:00
Pierre Albin: specified at more than 1 point.
12:12
Pierre Albin: and we say
12:17
Pierre Albin: we have Pierre Albin: a boundary value problem.
12:35
Pierre Albin: Another classification.
12:46
Pierre Albin: differential equations.
12:55
Pierre Albin: is into linear
13:00
Pierre Albin: and nonlinear.
13:10
Pierre Albin: a differential equation. It's linear
13:17
Pierre Albin: if
13:20
Pierre Albin: the dependent variable.
13:24
Pierre Albin: So Y, in the previous example.
13:34
Pierre Albin: all of its derivatives
13:43
Pierre Albin: enter into the equation linearly.
13:57
Pierre Albin: the dependence
14:04
Pierre Albin: on the independent, variable
14:10
Pierre Albin: right. So T, in the previous example.
14:16
Pierre Albin: does not have to be linear.
14:25
Pierre Albin: So, for example.
14:32
Pierre Albin: if we had an equation like second derivative of U, with respect to x Pierre Albin: plus E to the x times. the third derivative of U with respect to y
14:47
Pierre Albin: equals sine of XY. Pierre Albin: Right. This is Pierre Albin: a linear
15:00
Pierre Albin: third order.
15:02
Pierre Albin: partial differential equation.
15:06
Pierre Albin: right? Because Pierre Albin: the derivatives of U enter into the equation linearly.
15:17
Pierre Albin: On the other hand.
15:21
Pierre Albin: if we had something like third derivative of y with respect to t
15:28
Pierre Albin: plus y raised to the 20 seventh power plus y times the derivative of y. With respect to t equal to 5,
15:39
Pierre Albin: then we would have a nonlinear.
15:50
Pierre Albin: Third Order ode. Pierre Albin: So here, this is a Pde, because we're taking derivatives both respect y, and with respect x, with and respect. Y. Whereas this is an ode, because all of the derivatives are with respect. T.
16:05
This one is nonlinear. Pierre Albin: This this term Pierre Albin: the Y enters nonlinearly also. This term the Y enters nonlinearly the Y and its derivatives.
16:18
Pierre Albin: But if we only had this term. Of course it would be linear. and this Pierre Albin: because it doesn't have anything to do with Y or its derivatives, doesn't matter what it is. Similarly, this one term.
16:35
Pierre Albin: okay.
16:38
Pierre Albin: we'll spend a lot of time studying linear equations.
16:43
Pierre Albin: both because they are simpler, but also because in real life, when you are given something nonlinear. Pierre Albin: Usually the thing you do first to try to understand it is, look at linear approximations of it. because those are much easier to understand, and they tell you a lot about the nonlinear.
17:03
Pierre Albin: It's easy to write down the general form of a linear Pierre Albin: ode.
17:10
Pierre Albin: So the general
17:17
Pierre Albin: nth order
17:21
Pierre Albin: linear vote E
17:28
Pierre Albin: test form. Well, we'd have some coefficient function Pierre Albin: depending on t times the n-th derivative of y. With respect to T,
17:41
Pierre Albin: then some other coefficient function.
17:44
Pierre Albin: multiplying the N minus one derivative.
17:50
Pierre Albin: and so on until we have some coefficient function multiplying y itself. and then some other function of T,
18:00
Pierre Albin: when you have, and a linear equation written in this form.
18:09
Pierre Albin: We say that this equation
18:17
Pierre Albin: is homogeneous.
18:22
Pierre Albin: If the right-hand side is equal to 0
18:29
Pierre Albin: and non-homogeneous
18:35
device.
18:42
Pierre Albin: it turns out that if you can solve the homogeneous equation, then you can solve the non-homogeneous equation. So usually, when we study these, we'll Pierre Albin: focus first on the homogeneous version Pierre Albin: and try to find those solutions.
19:00
Pierre Albin: Okay.
19:02
Pierre Albin: so that's as far as we want to go and classify and differentiate equations for the moment. Let's talk about some examples.
19:27
Pierre Albin: Well, for first, example. Pierre Albin: if why represents, say, the position
19:41
Pierre Albin: at time T,
19:45
Pierre Albin: then, of course, Dy, Dt
19:49
Pierre Albin: represents the velocity
19:58
Pierre Albin: at time T,
20:01
Pierre Albin: and the second derivative of y with respect to T,
20:07
Pierre Albin: the acceleration.
20:13
Pierre Albin: So, for example. if we set up a simple equation like second derivative Pierre Albin: of y, with respect to T equal to minus g. When we think of an object falling Pierre Albin: solely under the force of gravity.
20:31
Pierre Albin: then Pierre Albin: well, this is a very simple equation to solve. You can just stare at it and see that the general solution would be Pierre Albin: minus g tai over 2 times t squared. Pierre Albin: plus some Pierre Albin: arbitrary
20:47
Pierre Albin: polynomial of degree at most one. and then you can find the constants. or you can find meaning for the constants.
20:57
Pierre Albin: C. One you could interpret
21:02
Pierre Albin: because it is dy. Dt. Pierre Albin: at time 0, this is just the initial velocity.
21:15
Pierre Albin: C, 0
21:18
Pierre Albin: is y had T equal to 0. This is just the initial height.
21:33
Pierre Albin: Okay, so for a second
21:37
Pierre Albin: set of examples, let's consider Pierre Albin: so Pierre Albin: growth right? So the derivative with respect to T, if we think of T as time is the rate of change. So Pierre Albin: we could be thinking about the growth of.
21:54
Pierre Albin: for example, Y could be some sort of population
22:03
Pierre Albin: at time. T.
22:06
Pierre Albin: Well, a very simple assumption to make about population growth Pierre Albin: is that the population should grow Pierre Albin: proportional to its size. Right?
22:21
Pierre Albin: The more people you have. the more Pierre Albin: the population will grow right and it grows proportional to Pierre Albin: size.
22:34
Pierre Albin: So a simple way to model that would be to say, Well, y prime is equal to some constant proportionality. Times y.
22:43
Pierre Albin: This, again is a very simple equation to solve. We can just write down a solution. right? For example, if if K. Were equal to one, then we'd be asking for Pierre Albin: a function whose derivative is that same function? Right? And of course the exponential Pierre Albin: is a function that does that
23:03
Pierre Albin: and
23:07
Pierre Albin: the general case y prime equal to k times y. The solution and said y of t is some constant times E to the k times, T.
23:19
So this Pierre Albin: exponential growth or exponential decay is perhaps not Pierre Albin: not too reasonable when you're thinking about, say, human population, because you eventually run out of resources. But this is really good.
23:41
Pierre Albin: We're modeling
23:44
Pierre Albin: radioactive decay.
23:53
Pierre Albin: Let's say, Y, equal to the mass of some material, some radioactive material.
24:07
Pierre Albin: Another simple model for growth is that you might think that the growth
24:16
Pierre Albin: should be proportional.
24:21
Pierre Albin: not necessarily to the size of the population you're looking at. but to the difference
24:32
Pierre Albin: with some external value.
24:38
Pierre Albin: So here, for example. we might say that T is the temperature
24:50
Pierre Albin: of somebody
24:53
Pierre Albin: at time. T.
24:56
Pierre Albin: And we might
24:58
Pierre Albin: want to compare that to what I'll call t naught the ambient temperature.
25:10
Pierre Albin: Right? So then, Newton's law of cooling
25:21
Pierre Albin: such that the change in temperature
25:25
Pierre Albin: is proportional
25:28
Pierre Albin: to the difference between the temperature and the ambient temperature. Pierre Albin: Right? So Pierre Albin: another very simple differential equation you can solve explicitly
25:42
Pierre Albin: might this. there's a first order ode.
25:48
Pierre Albin: It's linear. Pierre Albin: And there is. Pierre Albin: as we would expect, one arbitrary constant
25:56
Pierre Albin: in the general solution. Pierre Albin: But as long as this K of proportionality is positive, then we see that the temperature Pierre Albin: would decay exponentially until it reaches the ambient temperature and then stay there. Pierre Albin: although, of course, it would take an infinite time in this model for it to actually achieve that ambient temperature.
26:23
Pierre Albin: Okay, a third thing you might do
26:27
Pierre Albin: is suppose you're interested in population growth. Pierre Albin: But you want to be more realistic. And suppose we don't have
26:43
Pierre Albin: unlimited resources.
26:52
Pierre Albin: So then, maybe what you want to consider
27:09
Pierre Albin: both the current population
27:17
Pierre Albin: call it P.
27:22
Pierre Albin: And the difference
27:31
Pierre Albin: with the carrying capacity.
27:39
Pierre Albin: as it's called, which I'll denote. P. Naught. Pierre Albin: So then a model is known as the logistic model
27:50
Pierre Albin: suggests that the change in the population
27:55
Pierre Albin: should be proportional
27:59
Pierre Albin: to both the current population and the difference Pierre Albin: times. The difference with the carrying capacity. Pierre Albin: But okay, this again, is a simple differential equation. We'll we'll talk soon about how to solve this.
28:17
Pierre Albin: But for the moment I'll just tell you Pierre Albin: the solution is
28:30
Pierre Albin: in general and a function like this. So Pierre Albin: this one is a first order differential equation, right? Because we're only taking one derivative, but it's nonlinear, because you can see here, if you were to multiply this out, you'd end up with a term with P squared.
28:48
Pierre Albin: I Pierre Albin: over here in the general solution. Notice that, as expected, when you have a first order, differential equation, the general solution has one arbitrary constant
28:59
Pierre Albin: that you can't figure out from the equation, you would have to figure it out from some initial conditions.
29:07
Pierre Albin: Okay. why is this called logistic? That's a strange word. You don't see in other contexts. Logistic is the word that the person who proposed this model, a Belgian mathematician. Pierre Albin: Pierre Francois Vera host. used in 1845, Pierre Albin: when he introduced it. It's a it is a Greek word. It means a computation.
29:34
and the reason he used it is he was using logarithmic Pierre Albin: to describe what we would call exponential growth. Pierre Albin: And so this was to to compare it, to contrast it, perhaps, with logarithmic use. Logistic. That's where that word comes from.
29:55
Pierre Albin: Okay. So this shows some of the
30:00
Pierre Albin: usefulness of differential equations because Pierre Albin: each model is talking about something. But the Pierre Albin: techniques of describing just the rate of growth and then figuring out what that implies about the function Pierre Albin: doesn't care whether you, you're thinking of your function as being the temperature of something, or you're thinking of your function as being a population. Right? It's it abstracts that away.
30:28
Pierre Albin: And just says, if you want some relation between the rate of growth and your original function, then this is what that implies.
30:37
Pierre Albin: Right? So these are some settings. We'll see others. As the semester goes on one closing thought.
30:46
Pierre Albin: I want to leave you with
30:52
Pierre Albin: is that while solving
31:02
Pierre Albin: a differential equation can be hard.
31:13
Pierre Albin: But checking, your solution
31:22
Pierre Albin: is usually not hard.
31:27
Pierre Albin: and so I encourage you to take advantage of that Pierre Albin: and check your solutions when you've solved an equation. So, for example, here, this equation might be Pierre Albin: a little hard, especially before we talk about how you do it to solve. Pierre Albin: But still it's an equation where? Where you're only taking one derivative. So to check that something is a solution, not you just have to differentiate this once and then plug in and do a little bit of algebra.
31:53
Pierre Albin: So keep that in mind, as the semester goes on.
31:59
Pierre Albin: Okay. So one thing I will do throughout the semester is a Pierre Albin: I like to talk about history and other
32:10
Pierre Albin: context topics about how Pierre Albin: things that are near near to what we're talking about. Pierre Albin: that, I think, are cool, and that you might enjoy hearing about.
32:24
So that's what I'm going to do now.
32:28
We'll talk a little bit about the historical context.
32:33
Pierre Albin: Specifically, let me tell you a little about Newton and Leibniz.
32:40
Pierre Albin: However, this is not Pierre Albin: part of the course, so feel free to stop watching the video now Pierre Albin: and ignore what I'm about to say.
32:53
Pierre Albin: Okay, so the historical context. well, the first and simplest differential equation
33:02
Pierre Albin: is to try to find
33:05
Pierre Albin: some function whose derivative is another function.
33:11
Pierre Albin: This is as simple a differential equation as you could hope to write down. Pierre Albin: And of course we know the solution Pierre Albin: right? The solution is well, just integrate both sides with respect to T,
33:26
Pierre Albin: right? So solving a differential equation of this form is, it's the same as like coming up with the notion of an integral, and noticing that the integral is the opposite of the derivative right, the inverse function to the derivative.
33:41
Pierre Albin: While this had many precedents, we now attribute the discovery of Calculus to Newton and Leibniz.
33:51
Pierre Albin: So Isaac Newton.
34:00
Pierre Albin: right born on Christmas Day.
34:06
Pierre Albin: 1642
34:09
Pierre Albin: lived until 1727. So little Asterix steric for 2 reasons. One is that that's the same year that Galileo died. Pierre Albin: and I've always found that very significant. It's like a a passing of the baton, or something from from one great astronomer to another. Pierre Albin: one great scientist to another.
34:34
Pierre Albin: but also it's an asterisk, because that it's 1642. With respect to the old calendar, the the Julian calendar. Once Pierre Albin: England caught up to the rest of Europe and switched over to the Gregorian calendar just a little after Newton died. The dates got shifted. And so, actually, with respect to the Gregorian calendar, Newton was born in 1,643.
35:00
Pierre Albin: Newton was an English mathematician, and one of the greatest scientists who's ever lived.
35:06
Pierre Albin: What Pierre Albin: some of us find comfort knowing is that he was not an outstanding student. Pierre Albin: It wasn't until after he finished his undergraduate degree that he started showing his genius. He was a student at Cambridge University in England.
35:26
and in 1665 Pierre Albin: he stayed on to do his master's, and and so, in fact, he spent all of his academic career in Cambridge in 1665, when he was working on his Master's degree. There was a pandemic Pierre Albin: alright something we know a bit about Pierre Albin: from personal experience. It wasn't covid, it was bubonic plagues.
35:50
Pierre Albin: and they closed the university for 2 years
35:55
Pierre Albin: to top it off. During that same period in 1,666 there was what's known as the Great Fire of London. So the the English were having really bad couple years
36:07
now, since they did not have zoom. Pierre Albin: Newton spent his time deep in thought. and had what is sometimes called his andos
36:23
Pierre Albin: mirabilis.
36:26
Pierre Albin: right? 1666, which is just a fancy way of saying an amazing year. During that year he developed his first draft of the theory of Calculus.
36:41
Pierre Albin: He made huge advances in optics. In fact, he he's the one who noticed that Pierre Albin: white light is not the absence of color. It is the a mixture of all of the colors.
36:57
and Pierre Albin: he worked on his theory of gravity that, in fact, he refined his theories sufficiently
37:07
Pierre Albin: to convince himself that earth's gravity holds the moon Pierre Albin: in motion
37:15
Pierre Albin: all of that before the age of 25,
37:20
Pierre Albin: though, while many of the pieces of what we consider calculus, we're put together by others. For example, Newton's own teacher. a guy called Isaac Barrow. had proved the geometric version of the fundamental theorem of Calculus.
37:36
Pierre Albin: It was Newton and Leibniz Pierre Albin: who developed and systematized the theory to the the extent that we now credit them with discovering calculus Pierre Albin: right
37:48
Pierre Albin: for Newton calculus was what allowed his calculations Pierre Albin: in his physical investigations. Right? So notice Pierre Albin: the the not coincidental
38:01
Pierre Albin: words calculus and calculations. Right? That's why we call it calculus calculus means stone or pebble in Greek. and it refers to using, say, an abacus to do computations
38:17
Pierre Albin: so calculus refer to anything you used to make calculations. And after Newton Leibniz. Pierre Albin: What they came up with was so useful for doing computations that we just call it now the calculus
38:37
for Newton. Pierre Albin: It allowed him, for example, to show that his law of universal gravitation Pierre Albin: implied that the planets move in elliptic orbits, and explained 3 laws Pierre Albin: that the German astronomer, Henness Kepler, had noticed through meticulous observations. So Kepler just had tons of data. and he
38:58
Pierre Albin: studied the data and noticed 3 things that always happen which he calls his loss. And Pierre Albin: then Newton was able to show that these weren't were implied by his, his model. So it gave the model a ton of grievance.
39:18
And Pierre Albin: It really showed how strong calculus was, although mutant, did not publish
39:29
Pierre Albin: the calculus. When he published his findings in in astronomy.
39:36
Pierre Albin: he Pierre Albin: he used calculus to discover his results, but not to present them.
39:44
Pierre Albin: Newton finished his studies at Cambridge, and then he ended up becoming a professor when his professor, Isaac Barrow. Pierre Albin: left to become chaplain to the King. This would be Charles the Second
39:57
Pierre Albin: in 1669, Pierre Albin: and Newton got promoted to Professor special type of Professor notice to Lucas and Chair
40:07
Pierre Albin: in 16 Pierre Albin: 85. Charles the Second was succeeded by his brother, James Ii. James Ii. Was a Catholic. and.
40:20
Pierre Albin: as you know, the the English monarchy had had problems with the Catholic Church. Pierre Albin: Alright this Pierre Albin: during the whole Pierre Albin: reign of Henry the Eighth, and then his kids.
40:35
So it was a big deal that James Ii. Was a Catholic.
40:41
Pierre Albin: Newton, and basically everybody at Cambridge was were were Protestant. Pierre Albin: and there was some controversy with James wanting to to install Catholic professors. Pierre Albin: and Newton became embroiled in some political defence of the University. and this led to him, being elected to Parliament. where he worked with the majority of Parliament to offer the crowd to somebody else.
41:09
Pierre Albin: So this this led to the what's known as the Glorious Revolution. and James Ii. Was replaced by his sister Mary. The second was Protestant, and her husband, William the Third.
41:22
Pierre Albin: Newton. Pierre Albin: ended up, leaving Cambridge a few years later to take up a Government position at the Mint, the Royal Mint. Pierre Albin: and essentially stop doing research. Pierre Albin: The last portion of his life was dominated by a controversy over who had invented calculus him Pierre Albin: or live minutes. So this is
41:47
Pierre Albin: Gottfried
41:49
Pierre Albin: Wilhelm.
41:54
Pierre Albin: Sorry
41:58
Pierre Albin: live Links.
42:00
Pierre Albin: who was around from 1646 to 1716. So I put the Vaughn in parentheses, because Vaughn is used to denote nobility.
42:13
Pierre Albin: and Leibniz was not noble, but he just at some point started calling himself Pierre Albin: Vaughn Liveets, and some people went along with it.
42:22
Blindness was a German genius Pierre Albin: who was interested in everything. He's famous, not just in the history of math, but he's really famous also in the history of philosophy.
42:34
Pierre Albin: He? He did a lot beyond inventing calculus. He worked a lot on binary, for example. He didn't invent it, but he developed a lot.
42:42
Pierre Albin: and he he thought it had a lot of metaphysical importance that you could set up all of the numbers using just 0 and one, he thought this showed that all you needed was 0, which is nothing, and then one which he thought represented God. So as long as you add 0 and God, then you could do everything. Pierre Albin: But anyway, he also constructed like a primitive mechanical calculator.
43:09
Pierre Albin: What was special about it compared to previous attempts? Was that he figure out how to get it to do multiplication and division, as well as addition and subtraction. And he worked a lot as a diplomat.
43:24
Pierre Albin: You know, as opposed to. Like Newton, who spent all his time, his academic time at a university, and worked as a professor.
43:31
Pierre Albin: Leibniz, in his study, tried to come up with like global systems that would encompass all of human knowledge. Pierre Albin: As a philosopher he's perhaps best known for his optimism. He claimed that the Pierre Albin: the world we live in is the best of all possible worlds.
43:52
Pierre Albin: So you'd think that this would be a really hard position to defend, because, you know, lots of bad things happen. Pierre Albin: His explanation for things like natural disasters was that well, sure bad things happen, but to avoid them Pierre Albin: God would have had to create the world with different physical laws, and that would have led to a worse world. Pierre Albin: And so
44:18
Pierre Albin: bad things happen. But nevertheless, what we have is the best of all possible worlds.
44:24
Pierre Albin: Leibniz didn't study Pierre Albin: math or natural sciences, it would have been them. He studied law and philosophy. He got a doctorate in philosophy. and when he finished he worked. or a German baron.
44:40
Von Boinberg.
44:43
Pierre Albin: he works for this guy as his secretary, his assistant, his librarian, his lawyer, his advisor, and a wandering diplomat
44:53
Pierre Albin: as part of that the Baron sent him to to France and later England as a peace mission. So back, then, Louis Xiv. Pierre Albin: Was King of France, and and was really into war. Pierre Albin: So he went as part of Peace mission to try to convince Louis the Fourteenth not to attack parts of Germany. Of course, Germany as as a country didn't exist back then Pierre Albin: parts of what are now Germany.
45:22
So he ended up in Paris in 1672, Pierre Albin: where he spent some of his free time, because, being a diplomat Pierre Albin: involved back then a lot of free time where you're waiting for. The King and his court to have time to talk to you. Pierre Albin: So he spent his free time studying mathematics with A, with the famous Dutch mathematician, Christian Huygens.
45:45
Pierre Albin: We'll mention Huygens again later. and some of the things he did some of the really cool math he did when thinking about how to construct clocks. Pierre Albin: but any case, at this point Huygens is important, because he was the person who taught Pierre Albin: Math to Leibniz.
46:04
Pierre Albin: During this time in Paris, Lyman started to develop his ideas about calculus. And as part of that he spent a lot of time thinking about good notation. This is in stark contrast to Newton. Pierre Albin: Newton used calculus, basically just for his own computations. And so he just used whatever notation he came up with at the time Pierre Albin: and didn't put too much thought into it. His notations were actually really paid pretty terrible. whereas linemen's was great in notation.
46:35
Pierre Albin: He also made a couple of trips to London, which is interesting because of the controversy that later arose about who invented calculus.
46:46
Pierre Albin: But he ended up returning to Germany in 1676, because the Baron that he was working for had died, and he needed to find somebody else to help pay the bills. He found work with the Duke of Hanover
47:01
Pierre Albin: and worked for them the rest of his life.
47:07
Pierre Albin: mostly as a librarian, although the Duke also had him work on things like Pierre Albin: genealogical trees. Right? Of course, that's that's a a hobby for a lot of people these days. But Pierre Albin: but when you're Pierre Albin: working with the nobility can be really important for for inheritance to know exactly who you're related to. Pierre Albin: So that was important work. But far from that.
47:35
Anyway, blindness published his theory of calculus in like 1,684 Differential Calculus and Integral Calculus. A couple of years later Pierre Albin: Newton's book. The Principia appeared a year after that, so 1687.
47:53
But he had written Pierre Albin: his theory of calculus what Newton called the Theory of fluxions Pierre Albin: in 1671 so many years before. Limous. However, although Newton wrote it down. He didn't get it published. It just Pierre Albin: got
48:09
Pierre Albin: circulated around by Newton's friends.
48:14
Pierre Albin: So it's now known that Newton and Leibniz developed their theories independently of each other. They both came up with calculus.
48:24
Pierre Albin: But at the time there's a long controversy with Newton claiming that Leibniz had plagiarized his ideas. and that these these Pierre Albin: friends who had been circulating the manuscript that somehow Leonid had had gotten a look at it, and had was just pretending to have come up with the ideas on his own. Newton at this point was Pierre Albin: head of the Royal Society in in England. Pierre Albin: and so he could.
48:55
Pierre Albin: He used that a lot to his advantage in in this controversy
49:01
this led to a big split between science in England and science in the rest of Europe, because English mathematicians Pierre Albin: took Newton side, and European mathematicians took Leibniz side and Pierre Albin: anyway. So it it's really sad that this happened, and that these 2 geniuses spent so much of their time Pierre Albin: embroiled in that controversy.
49:29
Let's talk a little about the notation that limits is very justly famous for Pierre Albin: so Linus introduced, among other things, the notation for the integral. This is because for limits the integral was like an infinite sum. Pierre Albin: and this is how blandness wrote his S.
49:53
Pierre Albin: And so what we have is just a fancy S standing for a sum that we use for the integral. Newton didn't think of the integral as a as an infinite sum, he, he thought, if the integral is just the the inverse of the derivative which might call an antiderivative
50:10
Pierre Albin: blindness, also introduced the notation we use for derivatives, like D, by Dt, DY. By dt.
50:19
and for that matter, notation. Pierre Albin: DY for derivative. Y
50:26
Pierre Albin: so these are great, and we use them all the time, and they're definitely better than than anything Newton came up with.
50:34
Pierre Albin: He also came up with some notation we don't use. for example. Pierre Albin: limits did not use the equal sign that we use instead of the equals sign he used like a pie which was meant to represent like a balance. Right? So you know the sort of thing where you have. Pierre Albin: you know something hanging over here and something hanging over here Pierre Albin: right? And so you're like balancing the 2 sides
51:05
of an expression. And if they're perfectly balanced, then Pierre Albin: then you have something like this. What limeness would do for an inequality right? So instead of Pierre Albin: so pi Pierre Albin: instead of
51:26
Pierre Albin: equals, sign. And then, if you did this, but say the left leg was longer. Then this Pierre Albin: meant that the expression on the left was larger than the expression on the right. So this what he would use for greater than. Pierre Albin: And of course.
51:46
Pierre Albin: if you want less than then, you make this leg longer.
51:54
Pierre Albin: So while that's really cute Pierre Albin: this would be a problem for people like me who have messy handwriting, or you might not know whether you meant to make that like longer than the other one. Pierre Albin: and it didn't catch on another thing he would do that didn't catch on is, instead of doing parentheses. Pierre Albin: he would just put a line over the expression that that would be in frenzy. So so instead of Pierre Albin: and say
52:25
Pierre Albin: I, if you wrote 2, 3, plus 5, and then put a parentheses like this, this would be what we would write as 2 parentheses, 3 plus 5
52:35
Pierre Albin: right? Pierre Albin: Also even the notation that did catch on we use it a little differently. For for limits, the the integral. He is sort of like how we do square root these days. Pierre Albin: right? So where we have the square root of some expression, say 49, or something, and we put all of it underneath that bar. That's what he would do
53:00
Pierre Albin: with the integral sign. So what we would write today as the integral of X. Dx. He would have written as something like this. right? So his S. Included some overbar. Pierre Albin: So the notation he used. We don't use it the same way he did. But still it's it's wonderful notation that. That's also very suggestive of of things.
53:26
Pierre Albin: So anyway, that's Newton Leibniz.
53:31
Pierre Albin: and I will see you all next time.
Here to help you learn
Sign in to keep a private conversation with your course tutor.
Ask me anything about the material…
Sign in to chat