Divergence and curl: The language of Maxwell's equations, fluid flow, and more

Summary

Este video explora el concepto de campos vectoriales, ilustrando cómo asociar un vector a cada punto en el espacio puede representar fenómenos físicos como la fluidos o campos magnéticos. Se introducen los términos divergencia y rotulación, explicando cómo la divergencia indica la tendencia de flujo hacia afuera o hacia adentro de un punto, mientras que la rotulación mide la rotación del flujo alrededor de ese punto. Además, se discuten aplicaciones en física, como en las ecuaciones de Maxwell, y en sistemas dinámicos, como la ecología de poblaciones. El vídeo también aborda la relación entre estas operaciones y los productos punto y vectorial, proporcionando una visión intuitiva de su significado en contextos variados.

Takeaways

• 🌀 Un campo vectorial es una asociación de un vector a cada punto en el espacio, representando magnitude y dirección.
• 🎨 Los vectores en un campo vectorial pueden representar velocidades de fluidos, fuerzas de gravedad o campos magnéticos.
• 🖌️ Al dibujar campos vectoriales, se suelen acortar los vectores largos para no entorpecer la visualización, usando colores para indicar magnitud.
• ⏲️ Los campos vectoriales en la física pueden variar con el tiempo, pero en este vídeo se enfoca en campos vectoriales estáticos.
• 🤔 La divergencia y la rotulación (curl) son conceptos importantes en los campos vectoriales, y su comprensión se ve mejorada al imaginarlos en el contexto de flujos de fluidos.
• 💧 La divergencia en un punto del plano indica la tendencia de un fluido imaginario a fluir hacia afuera o hacia dentro de una región pequeña cercana a ese punto.
• 🔄 La rotulación (curl) mide la tendencia de rotación del fluido alrededor de un punto, con rotaciones de sentido horario teniendo curl positivo y contrahorario teniendo curl negativo.
• 🧲 Divergencia y curl son conceptos fundamentales en las ecuaciones de Maxwell, que describen la electricidad y la magnetismo.
• 🔄 La divergencia de un campo vectorial se relaciona con el producto punto, y la rotulación con el producto cruz, más allá de ser una herramienta de notación.
• 🔍 Estos conceptos matemáticos también son útiles en contextos no espaciales, como en la modelación de sistemas dinámicos en ecuaciones diferenciales.

Q & A

¿Qué es un campo vectorial?

Un campo vectorial es una asignación de un vector a cada punto en el espacio, representando magnitude y dirección, como podrían ser las velocidades de partículas de fluido o la fuerza de gravedad en diferentes puntos.

¿Por qué se acortan los vectores en los dibujos de campos vectoriales?

Se acortan los vectores en los dibujos para evitar que los vectores más largos ensombrezcan el diagrama, utilizando a menudo el color para dar una idea aproximativa de la longitud.

¿Cómo pueden los campos vectoriales cambiar con el tiempo?

Los campos vectoriales en la física pueden cambiar con el tiempo en respuesta al contexto circundante, como los cambios en la velocidad del viento o en los campos eléctricos a medida que las partículas cargadas se mueven.

¿Qué es la divergencia en un campo vectorial?

La divergencia de un campo vectorial en un punto específico indica cuánto fluido imaginario tiende a fluir hacia afuera o hacia dentro de una pequeña región cercana a ese punto.

¿Qué implica un valor positivo en la divergencia de un campo vectorial?

Un valor positivo en la divergencia indica que hay más fluido fluyendo hacia afuera de la región pequeña que entrando en ella, lo que sugiere una generación espontánea de fluido.

¿Qué es el giro o 'curl' en un campo vectorial?

El 'curl' de un campo vectorial en un punto dada indica cuánto el fluido tiende a girar alrededor de ese punto, como si se dejara caer una ramita en el fluido y se fijara su centro en su lugar.

¿Cómo se relaciona la divergencia con la función derivada?

La divergencia es análoga a una derivada, ya que da como resultado una nueva función que toma un punto en 2D como entrada y su salida depende del comportamiento del campo en una pequeña región alrededor de ese punto.

¿Por qué es importante la divergencia cero para los fluidos incompressibles?

Para un fluido incompressible, como el agua, la divergencia del campo vectorial debe ser cero en todas partes, lo que es una restricción importante para resolver problemas de fluidos reales.

¿Cómo se relacionan las ecuaciones de Maxwell con la divergencia y el 'curl'?

Las ecuaciones de Maxwell, fundamentales en la teoría electromagnética, están escritas en el lenguaje de divergencia y 'curl', conectando la divergencia del campo eléctrico con la densidad de carga y la divergencia del campo magnético con la ausencia de monopolos magnéticos.

¿Cómo pueden ser útiles la divergencia y el 'curl' en contextos que no son de flujo?

La divergencia y el 'curl' son útiles en contextos más allá de los flujos, como en la descripción de sistemas dinámicos en espacios de fase, donde pueden ayudar a entender la evolución de estados iniciales a través del tiempo.

Outlines

00:00

🌀 Introducción a los campos vectoriales y divergencia

El primer párrafo introduce los conceptos de campo vectorial, que asocia un vector con cada punto en el espacio, representando magnitude y dirección. Se menciona que estos vectores pueden simbolizar velocidades de fluidos, fuerzas de gravedad o campos magnéticos. Se discute la convención de dibujo de estos campos, donde se acortan los vectores largos para evitar el desorden visual y se utiliza el color para indicar la magnitud. Además, se explica que los campos vectoriales en la física pueden variar con el tiempo, pero en este vídeo se enfocarán en campos vectoriales estáticos en dos dimensiones. Se sugiere que visualizar un campo vectorial como un flujo de fluido puede ayudar a entender mejor propiedades como la divergencia y el giro, incluso si el campo representa otra cosa, como un campo eléctrico. La divergencia en un punto del plano indica la tendencia del flujo imaginario a salir de o entrar en una pequeña región cercana a ese punto, siendo positiva en fuentes y negativa en sumideros. Se destaca la importancia de la divergencia en la descripción de fluidos incompressibles, donde la divergencia debe ser cero en todos los puntos.

05:01

🔄 Concepto de giro (curl) y su aplicación en física

El segundo párrafo explora el concepto de giro (curl) en el contexto de flujos de fluidos, preguntando cuánto tiende a girar el fluido alrededor de un punto si se libera una partícula en él. Se describe cómo las regiones con giro de sentido horario tienen curl positivo y las de sentido antihorario, negativo. Aunque el giro es una idea tridimensional, en este vídeo se utiliza su variante bidimensional, asociando cada punto en el espacio 2D con un número en lugar de un vector. Se menciona que estos conceptos son importantes no solo para fluidos sino también para otros campos vectoriales, como los campos eléctricos y magnéticos, descritos por las ecuaciones de Maxwell. Estas ecuaciones relacionan la divergencia y el giro con la densidad de carga y la ausencia de monopolos magnéticos, respectivamente. Además, se sugiere que la comprensión de estos conceptos puede ser útil en contextos más allá de la física, como en la modelación de sistemas dinámicos en espacios de fase.

10:02

🔢 Importancia de la divergencia y el giro en ecuaciones diferenciales

El tercer párrafo enfatiza la utilidad de la divergencia y el giro en la comprensión de sistemas dinámicos más allá de la física clásica. Se da un ejemplo de cómo las poblaciones de especies, como depredadores y presas, pueden modelarse en un espacio de fase bidimensional, donde cada punto representa un estado del sistema en un momento dado. Se asocia cada punto con un vector que indica las tasas de cambio de las poblaciones, y se sugiere que visualizar estas tasas de cambio como un campo vectorial puede ayudar a entender cómo evolucionan los estados iniciales a lo largo del tiempo. Aunque se reconoce que para obtener una comprensión completa de estos sistemas se requieren herramientas matemáticas adicionales, la práctica con la divergencia y el giro proporciona una base útil para estudiar configuraciones similares. Se menciona la notación común para la divergencia y el giro, que se relaciona con el producto punto y el producto cruz, respectivamente, y se sugiere que estos operadores tienen una conexión real con las ideas geométricas subyacentes, más allá de ser simplemente una ayuda para recordar las formulas.

15:02

📢 Reflexiones sobre el patrocinio y la relación con el público

El cuarto párrafo se desvía del tema principal para discutir la decisión del canal de reducir el contenido patrocinado y enfocarse en una relación directa con el público a través de Patreon. Se argumenta que esta transición permite una mayor calidad y valía en el contenido, ya que las motivaciones se alinean con satisfacer directamente a los espectadores en lugar de maximizar el alcance. Se menciona que, aunque siempre se ha buscado aliarse con empresas que se alinean con los valores del canal, la ausencia de publicidad permite una mayor libertad para enfocarse en la calidad del contenido educativo. Se agradece a los patrocinadores existentes y se invita a los espectadores a explorar más sobre esta decisión en una publicación en Patreon.

Keywords

💡Vector field

Un campo vectorial es una asignación de un vector a cada punto en el espacio, donde cada vector tiene una magnitud y dirección. En el vídeo, los campos vectoriales representan conceptos como las velocidades de partículas en un fluido, la fuerza de gravedad o la intensidad de un campo magnético. El campo vectorial es fundamental para entender las propiedades y el comportamiento de estos fenómenos físicos, y es central para el tema del vídeo.

💡Divergence

La divergencia es una medida de la tendencia de un fluido imaginario a fluir hacia afuera o hacia dentro de una región pequeña cercana a un punto en particular. En el contexto del vídeo, la divergencia de un campo vectorial en un punto dado indica si hay una fuente o un sumidero en ese punto, y es una herramienta para entender la distribución y el flujo de un campo vectorial, como en el caso de los campos eléctricos.

💡Curl

El giro o curl mide la tendencia de un fluido a girar alrededor de un punto en un campo vectorial. En el vídeo, se utiliza el ejemplo de dejar caer una rama en un fluido y ver si tiende a girar, lo que indica la presencia de un giro. El curl es crucial para entender la rotación en sistemas físicos y es un concepto clave en la descripción de fenómenos como el flujo de fluidos y campos magnéticos.

💡Fluid flow

El flujo de fluidos es una metáfora utilizada en el vídeo para ilustrar conceptos matemáticos como la divergencia y el curl. Aunque no se trata de fluidos reales, el flujo de fluidos ayuda a visualizar cómo los campos vectoriales pueden representar fuerzas y movimientos en el espacio, facilitando la comprensión de conceptos abstractos.

💡Static vector fields

Los campos vectoriales estáticos son aquellos que no cambian con el tiempo, en contraposición a los campos vectoriales dinámicos que varían. En el vídeo, se centran en campos vectoriales estáticos para describir sistemas en estado estacionario, facilitando el estudio de propiedades como la divergencia y el curl sin la complejidad del tiempo variable.

💡Two-dimensional

A lo largo del vídeo, se enfatiza el estudio de campos vectoriales en dos dimensiones en lugar de tres. Esto simplifica la visualización y el análisis de los campos, permitiendo un enfoque más concentrado en las propiedades fundamentales de los campos vectoriales y sus aplicaciones en contextos como la física y la matemática.

💡Maxwell's equations

Las ecuaciones de Maxwell son cuatro ecuaciones que describen la interacción entre la electricidad y el magnetismo. En el vídeo, se menciona cómo la divergencia y el curl son fundamentales en estas ecuaciones, que son esenciales para entender fenómenos como la luz y las ondas electromagnéticas.

💡Gauss's law

La ley de Gauss es una de las ecuaciones de Maxwell que relaciona la divergencia de un campo eléctrico con la densidad de carga en un punto. En el vídeo, se utiliza para ilustrar cómo la divergencia puede interpretarse como una fuente o un sumidero de un fluido imaginario, proporcionando una visión intuitiva de la ley.

💡Differential equations

Las ecuaciones diferenciales son herramientas matemáticas utilizadas para describir cómo las cantidades cambian con el tiempo o con otras variables. En el vídeo, se menciona cómo las ecuaciones diferenciales pueden representarse en un campo vectorial, donde cada punto indica las tasas de cambio para una variable, lo que es útil para visualizar y entender dinámicas complejas.

💡Phase space

El espacio de fase es una representación en la que cada punto corresponde a un estado posible de un sistema dinámico. En el vídeo, se utiliza el espacio de fase para visualizar cómo las poblaciones de especies en un ecosistema pueden cambiar con el tiempo, mostrando cómo los campos vectoriales pueden aplicarse a contextos más allá de la física clásica.

Highlights

Vector fields associate each point in space with a vector, representing properties like magnitude and direction.

Vector fields can represent physical phenomena such as fluid velocities, gravitational forces, or magnetic field strengths.

In drawing vector fields, it's common to shorten long vectors to avoid clutter, using color to indicate magnitude.

Vector fields in physics can change over time, reflecting dynamic systems like fluid flow or electric fields.

This discussion focuses on static vector fields, which describe steady-state systems.

Vector fields are often three-dimensional, but this discussion is limited to two dimensions for simplicity.

Understanding one vector field by imagining it represents a different physical phenomenon can provide insights.

Divergence of a vector field indicates the tendency of an imagined fluid to flow out of or into a region.

Positive divergence at a point suggests a source of fluid, while negative divergence indicates a sink.

Curl of a vector field measures the rotation of fluid around a point, with positive and negative values indicating direction of rotation.

In real-world applications, such as incompressible fluids, the divergence of the velocity field must be zero.

Maxwell's equations, fundamental in electromagnetism, are formulated using divergence and curl.

Gauss's law relates the divergence of an electric field to the charge density at a point.

The divergence of a magnetic field being zero implies the absence of magnetic monopoles.

Divergence and curl are also relevant in non-spatial contexts, such as population dynamics in differential equations.

The phase flow of a system can be visualized as a vector field in the phase space of a dynamic system.

Divergent thinking with divergence and curl can inform about system behaviors like convergence or cyclic patterns.

The channel aims to focus on direct relationships with the audience, moving away from sponsored content.

The Patreon model is highlighted as a way to support content creation without relying on advertising.

Transcripts

Today, you and I are going to get into divergence and curl.

To make sure we're all on the same page, let's begin by talking about vector fields.

Essentially a vector field is what you get if you associate

each point in space with a vector, some magnitude and direction.

Maybe those vectors represent the velocities of particles of fluid

at each point in space, or maybe they represent the force of gravity

at many different points in space, or maybe a magnetic field strength.

Quick note on drawing these, often if you were to draw the vectors to scale,

the longer ones end up just cluttering up the whole thing,

so it's common to basically lie a little and artificially shorten ones that are too long,

maybe using color to give some vague sense of length.

In principle, vector fields in physics might change over time.

In almost all real-world fluid flow, the velocities of particles in a given

region of space will change over time in response to the surrounding context.

Wind is not a constant, it comes in gusts.

An electric field changes as the charged particles characterizing it move around.

But here we'll just be looking at static vector fields,

which maybe you think of as describing a steady-state system.

Also, while such vectors could in principle be three-dimensional,

or even higher, we're just going to be looking at two dimensions.

An important idea which regularly goes unsaid is that you can often

understand a vector field which represents one physical phenomenon

better by imagining what if it represented a different physical phenomenon.

What if these vectors describing gravitational force instead defined a fluid flow?

What would that flow look like?

And what can the properties of that flow tell us about the original gravitational force?

And what if the vectors defining a fluid flow were thought

of as describing the downhill direction of a certain hill?

Does such a hill even exist?

And if so, what does it tell us about the original flow?

These sorts of questions can be surprisingly helpful.

For example, the ideas of divergence and curl are particularly viscerally understood

when the vector field is thought of as representing fluid flow,

even if the field you're looking at is really meant to describe something else,

like an electric field.

Here, take a look at this vector field, and think of each

vector as describing the velocity of a fluid at that point.

Notice that when you do this, that fluid behaves in a very strange, non-physical way.

Around some points, like these ones, the fluid seems to just spring

into existence from nothingness, as if there's some kind of source there.

Some other points act more like sinks, where the

fluid seems to disappear into nothingness.

The divergence of a vector field at a particular point of the plane tells you

how much this imagined fluid tends to flow out of or into small regions near it.

For example, the divergence of our vector field evaluated at all

of those points that act like sources will give a positive number.

And it doesn't just have to be that all of the fluid is flowing away from that point.

The divergence would also be positive if it was just that the fluid coming

into it from one direction was slower than the flow coming out of it in another

direction, since that would still insinuate a certain spontaneous generation.

Now on the flip side, if in a small region around a point there seems to be more fluid

flowing into it than out of it, the divergence at that point would be a negative number.

Remember, this vector field is really a function that takes

in 2-dimensional inputs and spits out 2-dimensional outputs.

The divergence of that vector field gives you a new function,

one that takes in a single 2d point as its input,

but its output depends on the behavior of the field in a small

neighborhood around that point.

In this way it's analogous to a derivative, and that output is just a single number,

measuring how much that point acts as a source or a sink.

I'm purposely delaying discussion of computations here,

the understanding for what it represents is more important.

Notice, this means that for an actual physical fluid,

like water rather than some imagined one used to illustrate an arbitrary vector field,

then if that fluid is incompressible, the velocity vector field must have a divergence

of zero everywhere.

That's an important constraint on what kinds of vector

fields could solve real-world fluid flow problems.

For the curl at a given point, you also think about the fluid flow around it,

but this time you ask how much that fluid tends to rotate around the point.

As in, if you were to drop a twig in the fluid at that point,

somehow fixing its center in place, would it tend to spin around?

Regions where that rotation is clockwise are said to have positive curl,

and regions where it's counterclockwise have negative curl.

It doesn't have to be that all the vectors around the input are

pointing counterclockwise, or all of them are pointing clockwise.

A point inside a region like this one, for example, would also have non-zero curl,

since the flow is slow at the bottom, but quick up top,

resulting in a net clockwise influence.

And really, true proper curl is a three-dimensional idea,

one where you associate each point in 2d space with a new vector characterizing

the rotation around that point, according to a certain right-hand rule,

and I have plenty of content from my time at Khan Academy describing this in more

detail if you want, but for our main purpose, I'll just be referring to the

two-dimensional variant of curl, which associates each point in 2d space with a

single number rather than a new vector.

As I said, even though these intuitions are given in the context of fluid flow,

both of these ideas are significant for other sorts of vector fields.

One very important example is how electricity and

magnetism are described by four special equations.

These are known as Maxwell's equations, and they're

written in the language of divergence and curl.

This top one, for example, is Gauss's law, stating that the divergence of an

electric field at a given point is proportional to the charge density at that point.

Unpacking the intuition for this, you might imagine positively charged regions

as acting like sources of some imagined fluid,

and negatively charged regions as being the sinks of that fluid,

and throughout parts of space where there is no charge,

the fluid would be flowing incompressibly, just like water.

Of course, there's not some literal electric fluid,

but it's a very useful and pretty way to read an equation like this.

Similarly, another important equation is that the divergence of the

magnetic field is zero everywhere, and you can understand that by

saying that if the field represents a fluid flow,

that fluid would be incompressible, with no sources and no sinks, it acts just like water.

This also has the interpretation that magnetic monopoles,

something that acts just like a north or south end of a magnet in isolation,

don't exist, there's nothing analogous to positive and negative charges in

an electric field.

Likewise, the last two equations tell us that the way one of

these fields changes depends on the curl of the other field.

And really, this is a purely three-dimensional idea,

and a little outside of our main focus here, but the point is that

divergence and curl arise in contexts that are unrelated to flow,

and side note, the back and forth from these last two equations is what

gives rise to light waves.

And quite often, these ideas are useful in contexts

which don't even seem spatial in nature at first.

To take a classic example that students of differential equations often study,

let's say that you wanted to track the population sizes of two different species,

where maybe one of them is a predator of another.

The state of this system at a given time, meaning the two population sizes,

could be thought of as a point in two-dimensional space,

what you would call the phase space of this system.

For a given pair of population sizes, these populations may be

inclined to change based on things like how reproductive are the two species,

or just how much does one of them enjoy eating the other one.

These rates of change would typically be written

analytically as a set of differential equations.

It's okay if you don't understand these particular equations,

I'm just throwing them up for those of you who are curious,

and because replacing variables with pictures makes me laugh a little bit.

But the relevance here is that a nice way to visualize what such a set of

equations is really saying is to associate each point on the plane,

each pair of population sizes, with a vector, indicating the rates of change

for both variables.

For example, when there are lots of foxes, but relatively few rabbits,

the number of foxes might tend to go down because of the constrained food supply,

and the number of rabbits might also tend to go down because they're getting

eaten by all of the foxes, potentially at a rate that's faster than they can reproduce.

So a given vector here is telling you how, and how quickly,

a given pair of population sizes tends to change.

Notice, this is a case where the vector field is not about physical space,

but instead it's a representation of a certain dynamic system that has two variables,

and how that system evolves over time.

This can maybe also give a sense for why mathematicians

care about studying the geometry of higher dimensions.

What if our system was tracking more than just two or three numbers?

The flow associated with this field is called the phase flow for

our differential equation, and it's a way to conceptualize,

at a glance, how many possible starting states would evolve over time.

Operations like divergence and curl can help to inform you about the system.

Do the population sizes converge towards a particular pair of numbers,

or are there some values they diverge away from?

Are there cyclic patterns, and are those cycles stable or unstable?

To be perfectly honest with you, for something like this you'd often want to bring

in related tools beyond just divergence and curl, those would give you the full story,

but the frame of mind that practice with these two ideas brings you carries over

well to studying setups like this with similar pieces of mathematical machinery.

If you really want to get a handle on these ideas,

you'd want to learn how to compute them and practice those computations,

and I'll leave links to where you can learn about this and practice if you want.

Again, I did some videos and articles and worked examples for Khan Academy on this

topic during my time there, so too much detail here will start to feel redundant for me.

But there is one thing worth bringing up, regarding

the notation associated with these computations.

Commonly, the divergence is written as a dot product between this upside-down triangle

thing and your vector field function, and the curl is written as a similar cross product.

Sometimes students are told that this is just a notational trick,

each computation involves a certain sum of certain derivatives,

and treating this upside-down triangle as if it was a vector of derivative operators can

be a helpful way to keep everything straight.

But it is actually more than just a mnemonic device,

there is a real connection between divergence and the dot product,

and between curl and the cross product.

Even though we won't be doing practice computations here,

I would like to give you at least some vague sense for how these four ideas are connected.

Imagine taking some small step from one point of your vector field to another.

The vector at this new point will likely be a little different from the one

at the first point, there will be some change to the function after that step,

which you might see by subtracting off your original vector from that new one.

And this kind of difference to your function over

small steps is what differential calculus is all about.

The dot product gives you a measure of how aligned two vectors are, right?

The dot product of your step vector with that difference vector it causes

tends to be positive in regions where the divergence is positive, and vice versa.

In fact, in some sense, the divergence is a sort of average value for

this dot product of a step with a change to the output it causes over

all possible step directions, assuming that things are rescaled appropriately.

I mean, think about it, if a step in some direction causes a change to that vector in

that same direction, this corresponds to a tendency for outward flow,

for positive divergence.

And on the flip side, if those dot products tend to be negative,

meaning the difference vector is pointing in the opposite direction from the step vector,

that corresponds with a tendency for inward flow, negative divergence.

Similarly, remember that the cross product is a sort of measure for how perpendicular

two vectors are, so the cross product of your step vector with the difference vector

it causes tends to be positive in regions where the curl is positive, and vice versa.

You might think of the curl as a sort of average

of this step vector difference vector cross product.

If a step in some direction corresponds to a change perpendicular to that step,

that corresponds to a tendency for flow rotation.

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