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Taking a quote from Steven Strogatz,

Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations.

Of course, this language is spoken well beyond the boundaries of physics as well,

and being able to speak it and read it adds a new color to how you view the world around you.

In the next few videos, I want to give a sort of tour of this topic.

The aim is to give a big picture view of what this piece of math is all about,

while at the same time being happy to dig into the details of specific examples as they come along.

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,

but not too much beyond that.

Differential equations arise whenever it's easier to describe change than absolute amounts.

It's easier to say why population sizes, for example, grow or shrink,

than it is to describe why they have the particular values they do at some point in time.

It may be easier to describe why your love for someone

is changing than why it happens to be where it is now.

In physics, more specifically Newtonian mechanics, motion is often described in terms of force, and force determines acceleration, which is a statement about change.

These equations come in two different flavors, ordinary differential equations, or ODEs, equations, involving functions with a single input, often thought of as time,

and partial differential equations, or PDEs, dealing with functions that have multiple inputs.

Partial differential equations are something we'll be looking at more closely in the next video.

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,

or the velocity of a fluid at every point in space.

Ordinary differential equations, our focus for now, involve only a finite collection of values changing with time.

And it doesn't have to be time per se, your one independent variable could be something else,

but things changing with time are the prototypical and most common example of differential equations.

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.

As a nice warm -up, Consider the trajectory of something you throw in the air.

The force of gravity near the surface of Earth causes things to accelerate downward at 9 .8 meters per second per second.

Now, unpack what that's really saying.

It means if you look at that object free from other forces and record its velocity at every second,

these velocity vectors will accrue an additional downward component of 9 .8 meters per second every second.

We call this constant g for gravity.

This is enough to give us an example of a differential equation, albeit a relatively simple one.

Focus on the y coordinate as a function of time.

Its derivative gives the vertical component of velocity, whose derivative in turn gives the vertical component of acceleration.

For compactness, let's write that first derivative as y dot and that second derivative as y double dot.

Our equation says that y double dot is equal to negative g, a simple constant.

This is one where we can solve by integrating, which is essentially working the question backwards.

First, to find velocity, you ask, what function has negative g as a derivative?

Well, it's negative g times t, or more specifically negative g t plus the initial velocity.

Notice, there's many functions with this particular derivative, so you have an extra degree of freedom which is determined by an initial condition.

Now, what function has this as a derivative?

Well, it turns out to be negative one -half g times t squared plus that initial velocity times t,

and again we're free to add an additional constant without changing the derivative

and that constant is determined by whatever the initial position is.

And there you go, we just solved a differential equation, figuring out what a function is based on information about its rate of change.

Things get more interesting when the forces acting on a body depend on where that body is.

For example, studying the motion of planets, stars, and moons, gravity can no longer be considered a constant.

Given two bodies, the The pole on one of them is in the direction of the other,

with a strength inversely proportional to the square of the distance between them.

As always, the rate of change of position is velocity, but now the rate of change of velocity is some function of position,

so you have this dance between two mutually interacting variables, reminiscent of the dance between the two moving bodies which they describe.

This is reflective of the fact that often in differential equations, the puzzles you face involve finding a function whose derivative and

or higher order derivatives are defined in terms of the function itself.

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.

Higher order differential equations would be ones involving third derivatives, fourth derivatives, and so on.

Puzzles with more intricate clues.

The sensation you get when really meditating on one of these equations is one of solving an infinite continuous jigsaw puzzle.

In a sense, you have to find infinitely many numbers, one for each point in time t.

But they're constrained by a very specific way that these values intertwine with their own rate of change,

and the rate of change of that rate of change.

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.

How does this angle theta that it makes with the vertical change as a function of time?

This is often given as an example in introductory physics classes of harmonic motion, meaning it oscillates like a sine wave.

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,

and G is the strength of gravity.

However, these formulas are actually lies, or rather, approximations which only work in the realm of small angles.

If you were to measure an actual pendulum, what you'd find is that as you pull it out farther,

the period is longer than what the high school physics formulas would suggest.

And when you pull it out really far, this value of theta plotted versus time doesn't even look like a sine wave anymore.

To understand what's really going on, first things first, let's set up the differential equation.

We'll measure the position of the pendulum's weight as a distance x along this arc,

and And if the angle theta that we care about is measured in radians, we can write x as L times theta,

where L is the length of the pendulum.

As usual, gravity pulls down with an acceleration of g, but because the pendulum constrains the motion of this mass,

we have to look at the component of this acceleration in the direction of motion.

A little geometry exercise for you is to show that this little angle here is the same as theta.

So the component of gravity in the direction of motion, opposite this angle, will be negative g times sine of theta.

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.

And this minus sign in the acceleration indicates that it's always pointed in the opposite direction from displacement.

So what we have is that the second derivative of x, the acceleration, is negative g times sine of theta.

As always, it's nice to do a quick gut check that our formula makes physical sense.

When theta is zero, sine of zero is zero, so there's no acceleration in the direction of movement.

When theta is 90 degrees, sine of theta is one, so the acceleration is the same as it would be for freefall.

Alright, that checks out.

And because x is L times theta, that means the second derivative of theta is negative g over L times sine of theta.

To be a little more realistic, let's add in a term to account for the air resistance,

which maybe we model as being proportional to the velocity.

We'll write this as negative mu times theta dot, where mu is some constant that encapsulates all of the air resistance

and friction and such that determines how quickly the pendulum loses energy.

Now this, my friends, is a particularly juicy differential equation.

It's not easy to solve, but it's not so hard that we can't reasonably get some meaningful understanding out of it.

At first glance, you might think that the sine function you see here relates to the sine wave pattern for the pendulum.

Ironically though, what you will eventually find is that the opposite is true.

The presence of the sine in this equation is precisely why real pendulums don't oscillate with a sine wave pattern.

If that sounds odd, consider the fact that here the sine function is taking theta as an input,

but in the approximate solution you might see in a physics class, theta itself is oscillating as the output of a sine function.

Clearly something fishy is afoot.

One thing that I like about this example is that even though it's comparatively simple,

it exposes an important truth about differential equations that you need to grapple with.

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.

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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.

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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.

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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,

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chaos theory, has exposed

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that there are further limits on how well we can use these systems for prediction with or without solutions.

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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.

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We've even built some good understanding for why this happens.

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The three -body problem, for example, is known to have seeds of chaos within it.

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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.

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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.

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Этот урок разговорной практики построен на видео «Differential equations, a tourist's guide». Чаще всего повторяются слова: theta, equation, pendulum, space, differential. В этом видео 287 предложений и 4581 слов для шедоуинга. Речь длится 26:50. Говорящий говорит в естественном темпе, около 171 слов в минуту, близком к повседневной речи. Только 78% слов входят в 3000 самых частых слов английского языка, поэтому лексика сложная.

Ключевая лексика этого видео

15 слов из видео, которые стоит выучить, с произношением и значением:

Слово	Произношение	Значение
equation существительное	/ɪˈkweɪ.ʒən/	уравне́ние
derivative существительное	/dɪˈɹɪvətɪv/	произво́дное
vector существительное	/ˈvɛktɚ/	ве́ктор
velocity существительное	/vəˈlɑsəti/	ско́рость
physics существительное	/ˈfɪz.ɪks/	фи́зика
acceleration существительное	/əkˌsɛl.əˈɹeɪ.ʃən/	ускоре́ние, разго́н
solve глагол	/sɒlv/	реша́ть, реши́ть
component существительное	/kəmˈpoʊ.nənt/	компоне́нт, составна́я часть
trajectory существительное	/tɹəˈd͡ʒɛktəɹi/	траекто́рия
involve глагол	/ɪnˈvɑlv/	каса́ться, косну́ться
gravity существительное	/ˈɡɹævɪti/	серьёзность, ва́жность
dimensional прилагательное	/daɪˈmɛn.ʃə.nəl/	-ме́рный
diagram существительное	/ˈdaɪ.ə.ɡɹæm/	план, диагра́мма
angle существительное	/ˈæŋ.ɡəl/	у́гол
mathematics существительное	/mæθ(.ə)ˈmæt.ɪks/	матема́тика

Фразовые глаголы, которые вы услышите

Слово	Значение
figure out глагол	разбира́ться, разобра́ться
write down глагол	запи́сывать, записа́ть
check out глагол	проверя́ть, прове́рить
come along глагол	сопровожда́ть, составля́ть компа́нию
go back глагол	возвраща́ться, возврати́ться
go down глагол	спуска́ться, спусти́ться
go over глагол	хорошо́ принима́ться
move forward глагол	продвига́ть, продви́нуть

Грамматика в этом видео

Конструкции, которые говорящий использует чаще всего, с точными словами из видео:

Конструкция	В видео
Пассивный залог be + причастие прошедшего времени — важно, что происходит, а не кто это делает	are always expressed · is spoken · is often described
Present Perfect have/has + причастие прошедшего времени — прошлое действие, важное сейчас	has come · you've probably never heard · have just been punted
Придаточные определительные who / which + предложение — уточнение о человеке или предмете	acceleration, which is · integrating, which is · freedom which is

Произношение, на которое стоит обратить внимание

Говорящий использует 40 сокращённых и редуцированных форм, например you're, we'll, we're. Произносите их коротко, так, как слышите.

Звуки «th»: mathematics /mæθ(.ə)ˈmæt.ɪks/, nevertheless /ˌnɛvɚðəˈlɛs/, theta /ˈθiːtə/
Звуки «sh» и «zh»: equation /ɪˈkweɪ.ʒən/, differential [dɪfəˈɹənʃəɫ], acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, affection /əˈfɛk.ʃən/
Длинные слова — следите за ударением: differential [dɪfəˈɹənʃəɫ], derivative /dɪˈɹɪvətɪv/, velocity /vəˈlɑsəti/, acceleration /əkˌsɛl.əˈɹeɪ.ʃən/, trajectory /tɹəˈd͡ʒɛktəɹi/

Звуки, трудные для русскоязычных:

/w/ — губы округлены, это не /v/: equation /ɪˈkweɪ.ʒən/, swing /ˈswɪŋ/, downward /ˈdaʊnwɚd/, backward /ˈbækwɚd/, inward /ˈɪnwəd/
/h/ — лёгкий выдох, а не русское «х»: hence /ˈhɛns/, humanity /hjuˈmænɪti/, happily /ˈhæp.ə.li/, heck /hɛk/
Звонкие согласные в конце слова — не оглушайте их: derivative /dɪˈɹɪvətɪv/, solve /sɒlv/, involve /ɪnˈvɑlv/, evolve /ɪˈvɑlv/, correspond /ˌkoɹəˈspɑnd/

Как заниматься с этим видео

Прослушайте всё видео один раз молча и выпишите незнакомые слова.
Начните со скорости 0,75×, повторяйте предложение за предложением и вернитесь к обычной скорости, когда станет легко.
Запишите себя и сравните с оригиналом, обращая внимание на такие слова, как equation, derivative, vector.

Что такое техника Shadowing?

Shadowing — это научно обоснованная техника изучения языка, изначально разработанная для подготовки профессиональных переводчиков и популяризированная полиглотом доктором Александром Аргуэльесом. Метод прост, но эффективен: вы слушаете аудио на английском от носителей языка и немедленно повторяете вслух — как тень, следующая за говорящим с задержкой в 1–2 секунды. В отличие от пассивного прослушивания или грамматических упражнений, Shadowing заставляет мозг и мышцы рта одновременно обрабатывать и воспроизводить реальные речевые паттерны. Исследования показывают, что это значительно улучшает точность произношения, интонацию, ритм, связную речь, понимание на слух и беглость речи — что делает его одним из самых эффективных методов для подготовки к IELTS Speaking и реального общения на английском.

Техника шедоуинга: читать полное пошаговое руководство →

Практика Shadowing: Differential equations, a tourist's guide | DE1 - Изучайте разговорный английский по видео

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