Pratique du Shadowing: Differential equations, a tourist's guide | DE1 - Apprendre l'anglais à l'oral avec la vidéo
Choisissez un mode d'entraînement
Chaque mode ouvre une salle d'entraînement en plein écran.
287 phrases
Si les phrases sont trop courtes ou trop longues, cliquez sur Edit pour les ajuster.
1
Taking a quote from Steven Strogatz,
2
Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations.
3
Of course, this language is spoken well beyond the boundaries of physics as well,
4
and being able to speak it and read it adds a new color to how you view the world around you.
5
In the next few videos, I want to give a sort of tour of this topic.
6
The aim is to give a big picture view of what this piece of math is all about,
7
while at the same time being happy to dig into the details of specific examples as they come along.
8
I'll be assuming you know the basics of calculus, like what derivatives and integrals are, and in later videos we'll need some basic linear algebra,
9
but not too much beyond that.
10
Differential equations arise whenever it's easier to describe change than absolute amounts.
11
It's easier to say why population sizes, for example, grow or shrink,
12
than it is to describe why they have the particular values they do at some point in time.
13
It may be easier to describe why your love for someone
14
is changing than why it happens to be where it is now.
15
In physics, more specifically Newtonian mechanics, motion is often described in terms of force, and force determines acceleration, which is a statement about change.
16
These equations come in two different flavors, ordinary differential equations, or ODEs, equations, involving functions with a single input, often thought of as time,
17
and partial differential equations, or PDEs, dealing with functions that have multiple inputs.
18
Partial differential equations are something we'll be looking at more closely in the next video.
19
You often think of them as involving a whole continuum of values changing with time, like the temperature at every point of a solid body,
20
or the velocity of a fluid at every point in space.
21
Ordinary differential equations, our focus for now, involve only a finite collection of values changing with time.
22
And it doesn't have to be time per se, your one independent variable could be something else,
23
but things changing with time are the prototypical and most common example of differential equations.
24
Physics offers a nice playground for us here, with simple examples to start with, and no shortage of intricacy and nuance as we delve deeper.
25
As a nice warm -up, Consider the trajectory of something you throw in the air.
26
The force of gravity near the surface of Earth causes things to accelerate downward at 9 .8 meters per second per second.
27
Now, unpack what that's really saying.
28
It means if you look at that object free from other forces and record its velocity at every second,
29
these velocity vectors will accrue an additional downward component of 9 .8 meters per second every second.
30
We call this constant g for gravity.
31
This is enough to give us an example of a differential equation, albeit a relatively simple one.
32
Focus on the y coordinate as a function of time.
33
Its derivative gives the vertical component of velocity, whose derivative in turn gives the vertical component of acceleration.
34
For compactness, let's write that first derivative as y dot and that second derivative as y double dot.
35
Our equation says that y double dot is equal to negative g, a simple constant.
36
This is one where we can solve by integrating, which is essentially working the question backwards.
37
First, to find velocity, you ask, what function has negative g as a derivative?
38
Well, it's negative g times t, or more specifically negative g t plus the initial velocity.
39
Notice, there's many functions with this particular derivative, so you have an extra degree of freedom which is determined by an initial condition.
40
Now, what function has this as a derivative?
41
Well, it turns out to be negative one -half g times t squared plus that initial velocity times t,
42
and again we're free to add an additional constant without changing the derivative
43
and that constant is determined by whatever the initial position is.
44
And there you go, we just solved a differential equation, figuring out what a function is based on information about its rate of change.
45
Things get more interesting when the forces acting on a body depend on where that body is.
46
For example, studying the motion of planets, stars, and moons, gravity can no longer be considered a constant.
47
Given two bodies, the The pole on one of them is in the direction of the other,
48
with a strength inversely proportional to the square of the distance between them.
49
As always, the rate of change of position is velocity, but now the rate of change of velocity is some function of position,
50
so you have this dance between two mutually interacting variables, reminiscent of the dance between the two moving bodies which they describe.
51
This is reflective of the fact that often in differential equations, the puzzles you face involve finding a function whose derivative and
52
or higher order derivatives are defined in terms of the function itself.
53
In physics, it's most common to work with second order differential equations, which means the highest derivative you find in this expression is a second derivative.
54
Higher order differential equations would be ones involving third derivatives, fourth derivatives, and so on.
55
Puzzles with more intricate clues.
56
The sensation you get when really meditating on one of these equations is one of solving an infinite continuous jigsaw puzzle.
57
In a sense, you have to find infinitely many numbers, one for each point in time t.
58
But they're constrained by a very specific way that these values intertwine with their own rate of change,
59
and the rate of change of that rate of change.
60
To get a feel for what studying these can look like, I want you to take some time digging into a deceptively simple example, a pendulum.
61
How does this angle theta that it makes with the vertical change as a function of time?
62
This is often given as an example in introductory physics classes of harmonic motion, meaning it oscillates like a sine wave.
63
More specifically, one with a period of 2 pi times the square root of L over G, where where L is the length of the pendulum,
64
and G is the strength of gravity.
65
However, these formulas are actually lies, or rather, approximations which only work in the realm of small angles.
66
If you were to measure an actual pendulum, what you'd find is that as you pull it out farther,
67
the period is longer than what the high school physics formulas would suggest.
68
And when you pull it out really far, this value of theta plotted versus time doesn't even look like a sine wave anymore.
69
To understand what's really going on, first things first, let's set up the differential equation.
70
We'll measure the position of the pendulum's weight as a distance x along this arc,
71
and And if the angle theta that we care about is measured in radians, we can write x as L times theta,
72
where L is the length of the pendulum.
73
As usual, gravity pulls down with an acceleration of g, but because the pendulum constrains the motion of this mass,
74
we have to look at the component of this acceleration in the direction of motion.
75
A little geometry exercise for you is to show that this little angle here is the same as theta.
76
So the component of gravity in the direction of motion, opposite this angle, will be negative g times sine of theta.
77
Here we're considering theta to be positive when the pendulum is swung to the right, and negative when it's swung to the left.
78
And this minus sign in the acceleration indicates that it's always pointed in the opposite direction from displacement.
79
So what we have is that the second derivative of x, the acceleration, is negative g times sine of theta.
80
As always, it's nice to do a quick gut check that our formula makes physical sense.
81
When theta is zero, sine of zero is zero, so there's no acceleration in the direction of movement.
82
When theta is 90 degrees, sine of theta is one, so the acceleration is the same as it would be for freefall.
83
Alright, that checks out.
84
And because x is L times theta, that means the second derivative of theta is negative g over L times sine of theta.
85
To be a little more realistic, let's add in a term to account for the air resistance,
86
which maybe we model as being proportional to the velocity.
87
We'll write this as negative mu times theta dot, where mu is some constant that encapsulates all of the air resistance
88
and friction and such that determines how quickly the pendulum loses energy.
89
Now this, my friends, is a particularly juicy differential equation.
90
It's not easy to solve, but it's not so hard that we can't reasonably get some meaningful understanding out of it.
91
At first glance, you might think that the sine function you see here relates to the sine wave pattern for the pendulum.
92
Ironically though, what you will eventually find is that the opposite is true.
93
The presence of the sine in this equation is precisely why real pendulums don't oscillate with a sine wave pattern.
94
If that sounds odd, consider the fact that here the sine function is taking theta as an input,
95
but in the approximate solution you might see in a physics class, theta itself is oscillating as the output of a sine function.
96
Clearly something fishy is afoot.
97
One thing that I like about this example is that even though it's comparatively simple,
98
it exposes an important truth about differential equations that you need to grapple with.
99
They're really freaking hard to solve.
100
In this case, if we remove that dampening term, we can just barely write down an analytic solution.
101
But it's hilariously complicated.
102
It involves all these functions you've probably never heard of, written in terms of integrals and weird inverse integral problems.
103
And when you step back, presumably the reason for finding a solution is to then be able to make computations
104
and to build an understanding for whatever dynamics you're studying.
105
In this case, those questions have just been punted off to figuring out how to compute
106
and more importantly understand these new functions.
107
And more often, like if we add back in that dampening term, there is not a known way to write down an exact analytic solution.
108
Well, I mean, for any hard problem, you could just define a new function to be the answer of that problem.
109
Heck, even name it after yourself if you want.
110
But again, that's pointless unless it leads you to being able to make computations and to build understanding.
111
So instead, in the study of differential equations, we often do a sort of short circuit and skip the actual solution part,
112
since it's unattainable, and go straight to building understanding and making computations from the equations alone.
113
Let me walk through what that might look like with the pendulum.
114
What do you hold in your head, or what visualization can you get some software to pull up for you to understand the many possible ways
115
that a pendulum governed by these laws might evolve depending on its starting conditions?
116
You might be tempted to try imagining the graph of theta versus t, and somehow interpreting how this slope,
117
the position, and the curvature all interrelate with each other.
118
However, what will turn out to be both easier
119
and more general is to start by visualizing all possible states in a two -dimensional plane.
120
What I mean by the state of the pendulum is that you can describe it with two numbers, the angle and the angular velocity.
121
You can freely change either one of those two values without necessarily changing the other, but the acceleration is purely a function of those two values.
122
So each point of this two -dimensional plane fully describes the pendulum at any given moment.
123
You might think of these as all possible initial conditions of that pendulum.
124
If you know the initial angle and the initial angular velocity, that's enough to predict how the system will evolve as time moves forward.
125
If you haven't worked with them before, these sorts of diagrams can take a little getting used to.
126
What you're looking at now, this inward spiral, is a fairly typical trajectory for our pendulum,
127
so take a moment to think carefully about what is being represented.
128
Notice how at the start, as theta decreases, theta dot, the y coordinate, gets more negative.
129
Which makes sense, because the pendulum moves faster in the leftward direction as it approaches as the bottom.
130
Keep in mind, even though the velocity vector on this pendulum is pointed to the left,
131
the value of that velocity is always being represented by the vertical component of our space.
132
It's important to remind yourself that this state space is an abstract thing,
133
and is distinct from the physical space where the pendulum itself lives and moves.
134
Since we're modeling this as losing some of its energy to air resistance,
135
this trajectory spirals inward, meaning the peak velocity and peak displacement each go down a bit with each swing.
136
Our point is, in a sense, attracted to the origin, where theta and theta dot both equal zero.
137
With a space, we can visualize a differential equation as a vector field.
138
Here, let me show you what I mean.
139
The pendulum state is a vector, theta, theta dot.
140
Maybe you think of that as an arrow from the origin, or maybe you think of it as a point.
141
What matters is that it has two coordinates, each a function of time.
142
Taking the derivative of that vector gives you its rate of change, the direction and speed that it will tend to move in this diagram.
143
That derivative is a new vector, theta dot, theta double dot, which we visualize as being attached to the relevant point in space.
144
Now take a moment to interpret what this is saying.
145
The first component for this rate of change vector is theta dot, which is also a coordinate in our space.
146
So the higher up we are in the diagram, the more the point tends to move to the right, and the lower we are,
147
the more it tends to move to the left.
148
The vertical component is theta double dot, which our differential equation lets us rewrite entirely in terms of theta and theta dot itself.
149
In other words, the first derivative of our state vector is some function of that vector itself,
150
with most of the intricacy tied up in that second coordinate.
151
Doing the same at all points of this space will show how that state tends to change from any position.
152
As is typical with vector fields, we artificially scale down the vectors when we draw them to prevent clutter, but use color to loosely indicate magnitude.
153
Notice, we've effectively broken up a single second order equation into a system of two first order equations.
154
You might even give theta dot a different name to emphasize that we're really thinking of two separate values,
155
intertwined via this mutual effect they have on one another's rate of change.
156
This is a common trick in the study of differential equations.
157
Instead of thinking about higher order changes of a single value, we often prefer to think of the first derivative of vector values.
158
In this form, we have a wonderful visual way to think about what solving the equation means.
159
As our system evolves from some initial state, our point in this space will move along some trajectory in such a way that at every moment,
160
the velocity of that point matches the vector from this field.
161
And again, keep in mind, this velocity is not the same thing as the physical velocity of the pendulum.
162
It's a more abstract rate of change, encoding the rates of change for both theta and theta -dot.
163
You might find it fun to pause for a moment
164
and think through what exactly some of these trajectory lines say about the possible ways the pendulum evolves from different starting conditions.
165
For example, in regions where theta dot is quite high,
166
the vectors guide the point to travel to the right quite a ways before settling down into an inward spiral.
167
This corresponds to a pendulum with a high enough initial velocity
168
that it fully rotates around several times before settling into a decaying back and forth.
169
Earth.
170
Having a little more fun, when I tweak this air resistance term,
171
mu, say increasing it, you can immediately see how this will result in trajectories that spiral inward faster,
172
which is to say the pendulum slows down faster.
173
That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context,
174
not knowing that they described a pendulum.
175
It's not obvious just looking at them that increasing this value of mu means
176
that the system as a whole tends towards some attracting state faster.
177
So getting some software to draw these vector fields for you
178
can be a great way to build an intuition for how they behave.
179
What's wonderful is that any system of ordinary differential equations can be described by a vector field like this,
180
so it's a very general way to get a feel for them.
181
Usually though, they have many more dimensions.
182
For example, consider the famous three -body problem,
183
which is to predict how three masses in three -dimensional space evolve if they act on each other with gravity,
184
and if you know their initial positions and velocities.
185
Each mass has three coordinates describing its position, and three more describing its momentum.
186
So the system has 18 degrees of freedom in total, and hence an 18 -dimensional space of possible states.
187
It's a bizarre thought, isn't it?
188
A single point meandering through an 18 -dimensional space that we cannot visualize,
189
obediently taking steps through time based on whatever vector it happens to be sitting on from moment to moment,
190
completely encoding the positions and the momenta of the three masses we see in ordinary, physical, 3D space.
191
In practice, by the way, you can reduce the number of dimensions here by taking advantage of the symmetries of your setup,
192
but the point that more degrees of freedom results in higher dimensional state spaces remains the same.
193
In math, we often call a space like this a phase space.
194
You'll hear me use that term broadly for spaces encoding all kinds of states of changing systems,
195
but you should know that in the context of physics, especially Hamiltonian mechanics, the term is often reserved for a more special case,
196
namely a space whose axes represent position and momentum.
197
So a physicist would agree that the 18 -dimensional space describing the three -body problem is a phase space,
198
but they might ask that we make a couple of modifications to our pendulum setup for it to properly deserve the term.
199
For those of you who just watched the block collision video, the planes we worked with there would happily be called phase spaces by math folk,
200
though a physicist might prefer other terminology.
201
Just know that the specific meaning may depend on your context.
202
It may seem like a simple idea, depending on how well indoctrinated you are to modern ways of thinking about math,
203
but it's worth keeping in mind that it took humanity quite a while to really embrace thinking of dynamics spatially like this,
204
especially when the dimensions get very large.
205
In his book
206
Chaos, the author James Glick describes phase space as, One of the most powerful inventions of modern science.
207
One reason it's powerful is
208
that you can ask questions not just about a single initial condition but about a whole spectrum of initial states.
209
The collection of all possible trajectories is reminiscent of a moving fluid, so we call it phase flow.
210
To take one example of why phase flow is a fruitful idea, consider the question of stability.
211
The origin of our space corresponds to the pendulum standing still, and so does this point over here, representing when the pendulum is perfectly balanced upright.
212
These are the so -called fixed points of our system, and one natural question to ask is whether or not they're stable.
213
That is, will tiny nudges to the system result in a state that tends back towards that fixed point, or away from it?
214
Physical intuition for the pendulum makes the answer here kind of obvious, but how would you think about stability just looking at the equations,
215
say if they arose in some completely different, less intuitive context?
216
We'll go over how to compute the answers to questions like this in following videos, and the intuition for the relevant computations are guided heavily by
217
the thought of looking at small regions in space around a fixed point
218
and asking whether the flow tends to contract or expand.
219
And speaking of attraction and stability, let's take a brief sidestep to talk about love.
220
The Strogatz quote
221
that I mentioned earlier comes from a whimsical column in the New York Times on the mathematics of modeling affection,
222
an example well worth pilfering to illustrate that we're not just talking about physics here.
223
Imagine you've been flirting with someone, But there's been some frustrating inconsistency to how mutual your affection seems.
224
And perhaps during a moment when you turn your attention towards physics to keep your mind off the romantic turmoil,
225
mulling over the broken -up pendulum equations, you suddenly understand the on -again -off -again dynamics of your flirtation.
226
You've noticed that your own affection tends to increase when your companion seems interested in you, but decrease when they seem colder.
227
That is, the rate of change for your love is proportional to their feelings for you.
228
But this sweetheart of yours is precisely the opposite, strangely attracted to you when you seem uninterested, but turned off once you seem too keen.
229
The phase space for these equations looks very similar to the center part of your pendulum diagram.
230
The two of you will go back and forth between affection and repulsion in an endless cycle.
231
A metaphor of pendulum swings in your feelings would not just be apt, but mathematically verified.
232
In fact, if your partner's feelings were further slowed when they feel themselves too in love, let's say out of a fear of being made vulnerable,
233
we'd have a term matching the friction in the pendulum, and you too would be destined to an inward spiral towards mutual ambivalence.
234
I hear wedding bells already.
235
The point is that two very different seeming laws of dynamics, one from physics involving a single variable, and another from, uh, chemistry with two variables,
236
actually have a very similar structure, easier to recognize when you're looking at the phase diagram.
237
Most notably, even though the equations are different, for example there's no sine function in the romance equations, the phase space exposes an underlying similarity nevertheless.
238
In other words, you're not just studying a pendulum right now.
239
Now, the tactics you develop to study one case have a tendency to transfer to many others.
240
Okay, so phase diagrams are a nice way to build understanding.
241
But what about actually computing the answer to our equation?
242
Well, one way to do this is to essentially simulate what the universe would do,
243
but using finite time steps instead of the infinitesimals and limits defining calculus.
244
The basic idea is that if you're at some point in this phase diagram,
245
take a step based on the vector that you're sitting on for a small time step, delta t.
246
Specifically, take a step equal to delta t times that vector.
247
As a reminder, in drawing these vector fields, the magnitude for each vector has been artificially scaled down to prevent clutter.
248
When you do this repeatedly, your final location will be an approximation of theta t, where t is the sum of all of those time steps.
249
If you think about what's being shown right now, though, and what that would imply for the pendulum's movement,
250
you'd probably agree that this is grossly inaccurate.
251
But that's only because the time step delta t of 0 .5 is way too big.
252
If we turned it down, say to 0 .01, you can get a much more accurate approximation.
253
It just takes more repeated steps is all.
254
In this case, computing theta of 10 requires 1 ,000 little steps.
255
Luckily, we live in a world with computers,
256
so repeating a simple task a thousand times is as simple as articulating that task with a programming language.
257
In fact, let's finish things off by writing a little
258
Python program that computes theta for us.
259
What it has to do is make use of the differential equation,
260
which returns the second derivative of theta as a function of theta and theta dot.
261
You start off by defining two variables, theta and theta dot, each in terms of some initial conditions.
262
In this case I'll have theta start at pi thirds, which is 60 degrees, and theta dot start at zero.
263
Next, write a loop that corresponds to taking many little time steps between zero and your time t, each of size delta t,
264
which I'm setting here to be 0 .01.
265
In each step of this loop, increase theta by theta dot times delta t, and increase theta dot by theta double dot times delta t,
266
where theta double dot can be computed based on the differential equation.
267
After all these little time steps, simply return the value of theta.
268
This is called solving a differential equation numerically.
269
Numerical methods can get way more sophisticated and intricate than this to better balance the tradeoff between accuracy and deficiency,
270
but this loop gives the basic idea.
271
So even though it sucks that we can't always find exact solutions,
272
there are still meaningful ways to study differential equations in the face of this inability.
273
In the following videos we will look at several methods for finding exact solutions when it's possible,
274
but one theme I'd like to focus on is how these
275
exact solutions can also help us to study the more general, unsolvable cases. But it gets worse.
276
Just as there's a limit to how far exact analytic solutions can get us, one of the great fields to have emerged in the last century,
277
chaos theory, has exposed
278
that there are further limits on how well we can use these systems for prediction with or without solutions.
279
Specifically, we know that for some systems, small variations to the initial conditions, say the kind due to necessarily imperfect measurements,
280
result in wildly different trajectories.
281
We've even built some good understanding for why this happens.
282
The three -body problem, for example, is known to have seeds of chaos within it.
283
So, looking back at the quote from earlier, it seems almost cruel of the universe to fill its language with riddles that we either can't solve,
284
or where we know that any solution would be useless for long -term prediction anyway.
285
It is cruel, but then again, it should also be reassuring.
286
It gives some hope that the complexity we see in the world around us can be studied somewhere in this math,
287
and that it's not hidden away in the mismatch between model and reality.
Télécharger l'application
Tout pour parler couramment
Prononciation IAChaque phrase notée
Phonétique (IPA)Maîtrisez chaque son
VocabulaireEnrichissez votre lexique
Jeu de vocabulaireApprenez en jouant
Même chaîne
C13
The Newton art puzzle
American AccentUniversity StudentsWorking Adults
C15
Transformers, the tech behind LLMs | Deep Learning Chapter 5
American AccentBusiness EnglishUniversity StudentsWorking Adults
C246
Large Language Models explained briefly
American AccentBusiness EnglishUniversity StudentsWorking Adults
C12
Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C15
Fourier series
American AccentUniversity StudentsWorking Adults
C14
Euler's Formula Poem
American AccentIELTS PrepTOEFL PrepUniversity Students
C12
Vectors | Chapter 1, Essence of linear algebra
American AccentBusiness EnglishUniversity StudentsWorking Adults
C13
How might LLMs store facts | Deep Learning Chapter 7
American AccentIELTS PrepTOEFL PrepUniversity Students
C113
Essence of linear algebra preview
American AccentBusiness EnglishUniversity StudentsWorking Adults
C15
The inscribed square problem
American AccentUniversity StudentsWorking Adults
C12
Oh, wait, actually the best Wordle opener is not “crane”…
American AccentBusiness EnglishUniversity StudentsWorking Adults
C21
Hilbert's Curve: Is infinite math useful?
American AccentBusiness EnglishUniversity StudentsWorking Adults
C26
But what is a neural network? | Deep learning chapter 1
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1Divergence and curl: The language of Maxwell's equations, fluid flow, and more
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1But what is quantum computing? (Grover's Algorithm)
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1Tie random ends: How many loops?
American AccentUniversity StudentsWorking Adults
FOR YOU
Vidéo recommandée
C13
279 phrases
LET THERE BE... Voltage? | Maxwell's Equation #2 Explained for Beginners
Daily ConversationUniversity Students
C11
51 phrases
What's Under Your Feet in New York City?
Daily ConversationUniversity StudentsWorking Adults
C1117 phrases
Anatomy & Physiology: The Ultimate Beginner's Guide | Dr. Bee's Tutorials
Daily ConversationUniversity Students
C11
362 phrases
How The Economic Machine Works by Ray Dalio
Daily ConversationUniversity StudentsWorking Adults
C11
296 phrases
John Berger / Ways of Seeing , Episode 1 (1972)
Daily ConversationUniversity Students
C11
244 phrases
1. Introduction to statistics - Lecture 1
Daily ConversationUniversity Students
C12
261 phrases
Your Life as Every Industrial Engineer Rank
Daily ConversationUniversity Students
C12
138 phrases
Derivatives: Crash Course Physics #2
Daily ConversationUniversity Students
C1552 phrases
The Professor Who Taught People How To Think (1962)
Daily ConversationUniversity StudentsWorking Adults
C11
154 phrases
How do some languages not have tense?
Daily ConversationUniversity Students
C13
119 phrases
Mathematics and Physics | Lesson 2 | Understanding the Connection
Daily ConversationUniversity Students
C11
498 phrases
Give me 31 minutes, I will make PyTorch click forever
Daily ConversationUniversity Students
Vocabulaire et conseils d’expression pour cette leçon
Cette leçon d’expression orale s’appuie sur la vidéo « Differential equations, a tourist's guide ». Les mots qui reviennent le plus souvent : theta, equation, pendulum, space, differential. Cette vidéo contient 287 phrases et 4581 mots à répéter en shadowing. La partie parlée dure 26:50. Le locuteur parle à un rythme naturel d’environ 171 mots par minute, proche d’une conversation courante. Seuls 78 % des mots font partie des 3 000 mots les plus courants en anglais, le vocabulaire est donc exigeant.
Vocabulaire clé de cette vidéo
15 mots de la vidéo à retenir, avec leur prononciation et leur sens :
| Mot | Prononciation | Sens |
|---|---|---|
| equation nom | /ɪˈkweɪ.ʒən/ | équation |
| derivative nom | /dɪˈɹɪvətɪv/ | dérivé |
| vector nom | /ˈvɛktɚ/ | vecteur |
| velocity nom | /vəˈlɑsəti/ | vecteur vitesse, vélocité |
| physics nom | /ˈfɪz.ɪks/ | physique |
| acceleration nom | /əkˌsɛl.əˈɹeɪ.ʃən/ | accélération |
| solve verbe | /sɒlv/ | résoudre, régler |
| component nom | /kəmˈpoʊ.nənt/ | composant |
| trajectory nom | /tɹəˈd͡ʒɛktəɹi/ | trajectoire |
| involve verbe | /ɪnˈvɑlv/ | impliquer, involver |
| gravity nom | /ˈɡɹævɪti/ | gravité |
| dimensional adjectif | /daɪˈmɛn.ʃə.nəl/ | dimensionnel, de dimension |
| diagram nom | /ˈdaɪ.ə.ɡɹæm/ | diagramme, schéma |
| angle nom | /ˈæŋ.ɡəl/ | angle |
| mathematics nom | /mæθ(.ə)ˈmæt.ɪks/ | mathématiques |
Les verbes à particule que vous entendrez
| Mot | Prononciation | Sens |
|---|---|---|
| break up verbe | rompre, couper | |
| figure out verbe | cerner | |
| write down verbe | noter, écrire | |
| check out verbe | mater, regarder | |
| come along verbe | accompagner, venir | |
| go back verbe | rentrer, retourner | |
| go down verbe | descendre | |
| set up verbe | /ˌsɛt ˈʌp/ | mettre en place, installer |
La grammaire de cette vidéo
Les structures que le locuteur utilise le plus, avec les mots exacts de la vidéo :
| Structure | Dans la vidéo |
|---|---|
| Voix passive be + participe passé — l’accent est mis sur ce qui arrive, pas sur qui le fait | are always expressed · is spoken · is often described |
| Present perfect have/has + participe passé — une action passée qui compte encore maintenant | has come · you've probably never heard · have just been punted |
| Propositions relatives who / which + proposition — une précision sur une personne ou une chose | acceleration, which is · integrating, which is · freedom which is |
Prononciation à surveiller
Le locuteur utilise 40 contractions et formes réduites, comme you're, we'll, we're. Prononcez-les sous leur forme courte, telles que vous les entendez.
- Les sons « th »: mathematics /mæθ(.ə)ˈmæt.ɪks/, nevertheless /ˌnɛvɚðəˈlɛs/, theta /ˈθiːtə/
- Les sons « sh » et « zh »: equation /ɪˈkweɪ.ʒən/, differential [dɪfəˈɹənʃəɫ], acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, affection /əˈfɛk.ʃən/
- Mots longs — placez bien l’accent: differential [dɪfəˈɹənʃəɫ], derivative /dɪˈɹɪvətɪv/, velocity /vəˈlɑsəti/, acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, trajectory /tɹəˈd͡ʒɛktəɹi/
Les sons difficiles pour les francophones :
- /h/ — il se souffle, il n’est pas muet: hence /ˈhɛns/, humanity /hjuˈmænɪti/, happily /ˈhæp.ə.li/, heck /hɛk/
- /tʃ/ et /dʒ/ — à ne pas adoucir en « ch » et « j »: trajectory /tɹəˈd͡ʒɛktəɹi/, mutual /ˈmjuːt͡ʃuəl/, emerge /ɪˈmɝd͡ʒ/, pendulum /ˈpɛnd͡ʒələm/
- /r/ anglais — langue recourbée, sans frotter la gorge: differential [dɪfəˈɹənʃəɫ], derivative /dɪˈɹɪvətɪv/, acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, trajectory /tɹəˈd͡ʒɛktəɹi/, gravity /ˈɡɹævɪti/
Comment s’entraîner avec cette vidéo
- Écoutez la vidéo en entier une fois sans parler et notez les mots que vous ne connaissez pas.
- Commencez à la vitesse 0,75×, répétez phrase par phrase, puis revenez à la vitesse normale quand cela devient facile.
- Enregistrez-vous et comparez avec l’original, en faisant attention à des mots comme equation, derivative, vector.
Qu'est-ce que la technique du Shadowing ?
Le Shadowing est une technique d'apprentissage des langues fondée sur la science, développée à l'origine pour la formation des interprètes professionnels. Le principe est simple mais puissant : vous écoutez de l'anglais natif et le répétez immédiatement à voix haute — comme une ombre suivant le locuteur avec un décalage de 1 à 2 secondes. Les recherches montrent une amélioration significative de la précision de la prononciation, de l'intonation, du rythme, des liaisons, de la compréhension orale et de la fluidité.