The Curl of a Vector Field: Measuring Rotation

Summary

This lecture delves into the concept of curl in vector calculus, explaining its role in measuring the rotation of vector fields. The presenter defines curl, illustrates its computation, and applies it to examples including fluid flows and solid body rotations. Key properties like the curl of a gradient being zero and the divergence of a curl also being zero are highlighted, setting the stage for further exploration into conservation laws and partial differential equations.

Takeaways

• 📚 The lecture discusses the concept of the curl in vector calculus, which measures the rotation in a vector field.
• 🔍 The curl is defined as the cross product of the del operator (∇) and a vector-valued function, represented in three dimensions.
• 🧩 The curl operation takes a vector field and outputs another vector field, unlike the gradient and divergence which transform between scalars and vectors.
• 🌀 The curl is computed using the determinant of a 3x3 matrix involving the del operator and the components of the vector field.
• 🌐 The script provides an example of computing the curl for a specific vector field, illustrating the algebra involved in the calculation.
• 💡 The interpretation of the curl is discussed, explaining its physical meaning in terms of rotation within a vector field.
• 🔄 The curl is used to identify rotational motion in fluid flows and solid body rotations, such as an asteroid spinning in space.
• 🌟 The script highlights that a vector field with zero curl is called curl-free or irrotational, indicating no rotational component.
• 🚫 Important properties of the curl are mentioned: the curl of a gradient is always zero, and the divergence of a curl is always zero.
• 🔍 The lecture concludes by connecting the concepts of gradient, divergence, and curl to the derivation of conservation laws and partial differential equations in physical systems.

Q & A

What are the three main vector calculus operations discussed in the script?

The three main vector calculus operations discussed in the script are the divergence, the gradient, and the curl.

What does the curl of a vector field represent?

The curl of a vector field represents the amount of rotation in the field. It is a measure of how much the field is rotating around a given point.

How is the curl defined mathematically?

The curl is defined as the cross product of the del operator (∇) with a vector-valued function f, mathematically represented as curl(f) = ∇ × f.

What is the difference between the gradient, divergence, and curl in terms of their outputs?

The gradient takes in a scalar and returns a vector, the divergence takes in a vector and returns a scalar, and the curl takes in a vector and returns a vector.

What is the physical interpretation of a vector field having a curl of zero?

A vector field with a curl of zero is considered curl-free or irrotational, meaning there is no rotational component in the field. This implies that particles within the field will not rotate around any point.

How does the script describe the computation of the curl for a 2D vector field?

For a 2D vector field, the curl is computed as the partial derivative of the second component with respect to the x-coordinate minus the partial derivative of the first component with respect to the y-coordinate, and it is always in the k (or z) direction.

What is the significance of the right-hand rule in the context of the curl?

The right-hand rule is used to determine the direction of the curl. If you curl your fingers from the x-axis to the y-axis, your thumb points in the direction of the positive z-axis, indicating the direction of the curl.

How does the script relate the concept of curl to fluid flow and solid body rotation?

The script explains that the curl can be used to describe the rotation of fluid flows, such as in the Gulf of Mexico or a bathtub, and also the rotation of solid bodies like asteroids, where the curl represents the angular velocity vector of the rotation.

What are the implications of the curl of a gradient being zero?

The curl of a gradient being zero implies that any potential flow vector field, such as the gravitational field on Earth, is irrotational. This means that the flow is coming in at a constant rate without any rotational component.

Why is the divergence of the curl always zero?

The divergence of the curl is always zero because the curl operation extracts the rotational part of a vector field, which inherently has no divergence. This means that the curl of any vector field is always a divergence-free vector field.

Outlines

00:00

📚 Introduction to Vector Calculus Operations

This paragraph introduces the concepts of divergence, gradient, and curl in vector calculus. The speaker begins by discussing the divergence and gradient from previous lectures and then transitions to the curl, emphasizing its importance in understanding the rotation of vector fields. Two vector fields are examined: one with radial symmetry and no rotation, and another with rotation but zero divergence. The speaker aims to define the curl, compute it for these examples, and explore its applications in fluid dynamics and solid body rotations.

05:01

🔍 Defining and Computing the Curl

The speaker defines the curl as a differential operator, specifically the cross product of the del operator (∇) with a vector-valued function. The curl is shown to take a multi-component vector field and output another vector field, distinguishing it from the gradient and divergence operators. An example vector field is given, and the speaker demonstrates how to compute the curl by calculating the determinant of a 3x3 matrix. The computation reveals the i, j, and k components of the curl, highlighting the process of converting between ijk notation and column vector notation.

10:05

🌀 Understanding the Physical Interpretation of Curl

The speaker delves into the physical interpretation of the curl, explaining that it represents the rotational aspect of a vector field. Two 2D vector fields are analyzed: one that is radially symmetric and expected to have zero curl, and another that is rotational and expected to have a non-zero curl. The computation of the curl for these fields is demonstrated, revealing that the first field is indeed curl-free, while the second has a curl of two. The right-hand rule is introduced to determine the direction of rotation, and the concept of solid body rotation is briefly introduced.

15:07

🚀 Applying Curl to Solid Body Rotations

The speaker explores the application of the curl in the context of solid body rotations, such as an asteroid spinning in space. The curl is used to describe the velocity of points on the asteroid due to its rotation. The angular velocity vector is defined, and the relationship between the rotation vector, the position vector, and the velocity vector is established through the cross product. The speaker illustrates how the curl can be used to determine the motion of every point on a rotating object, emphasizing the importance of the curl in understanding rotational dynamics.

20:08

🔄 Further Insights into Curl and Its Properties

The speaker provides further insights into the curl, explaining that the curl of a gradient is always zero, indicating that any potential flow vector field is irrotational. Examples such as the gravitational vector field on Earth are given to illustrate this point. Additionally, the divergence of a curl is shown to always be zero, highlighting that the curl isolates the rotational component of a vector field. These properties are emphasized as important in understanding the behavior of vector fields in physical systems.

25:09

📘 Conclusion and Preview of Future Topics

In the concluding paragraph, the speaker summarizes the discussion on the gradient, divergence, and curl, setting the stage for future topics. The focus shifts to the application of these concepts in writing conservation laws and deriving partial differential equations for real physical systems. The speaker emphasizes the importance of these vector calculus operations in understanding and modeling physical phenomena.

Keywords

💡Divergence

Divergence is a vector calculus operation that measures the rate at which a vector field spreads out or converges at a given point. In the video, divergence is discussed in the context of vector fields, where it is used to determine how much a vector field is diverging or converging. For example, the script mentions a vector field that is diverging and another with divergence zero, illustrating the concept through visual examples.

💡Gradient

The gradient is a vector calculus operation that indicates the direction and rate of the fastest increase of a scalar field. In the video, the gradient is mentioned in the context of the last two lectures, where it is used to describe how changes in a scalar field can be represented as a vector field. The script also discusses the relationship between the gradient and the curl, emphasizing their distinct roles in vector calculus.

💡Curl

Curl is a vector calculus operation that measures the rotation or circulation of a vector field. The video focuses on the curl, explaining how it can be used to determine the rotational component of a vector field. The script provides examples of vector fields with zero curl and non-zero curl, demonstrating the concept through visual illustrations and mathematical computations.

💡Vector Field

A vector field is a mathematical concept that assigns a vector to every point in space. In the video, vector fields are central to the discussion of divergence, gradient, and curl. The script uses vector fields to illustrate how these operations can be applied to understand the behavior of physical phenomena like fluid flow and solid body rotations.

💡Nabla

Nabla, often denoted by the symbol 'del', is a differential operator used in vector calculus. In the video, nabla is introduced as the operator used in the cross product with a vector field to compute the curl. The script explains how nabla operates in three dimensions, highlighting its role in the mathematical formulation of the curl.

💡Cross Product

The cross product is a binary operation on two vectors in three-dimensional space, resulting in a vector that is perpendicular to the plane containing the input vectors. In the video, the cross product is used to define the curl of a vector field, with the script detailing how the cross product of nabla and a vector function can be computed using determinants.

💡Scalar

A scalar is a mathematical entity that can be described with a single number, representing a quantity without direction. In the video, scalars are contrasted with vectors, with the script explaining how the gradient operation takes a scalar field and returns a vector field, indicating the direction of the steepest ascent.

💡Solid Body Rotation

Solid body rotation refers to the rotation of an object as a whole, with all parts of the object moving in concentric circles around the axis of rotation. In the video, solid body rotation is used to illustrate the physical interpretation of curl, with the script explaining how the curl of a vector field associated with solid body rotation is constant and related to the angular velocity of the rotation.

💡Irrotational

Irrotational is a term used to describe a vector field that has zero curl everywhere, indicating no rotational motion. In the video, the script discusses the concept of irrotational vector fields, providing examples and explaining how such fields can be associated with conservative forces like gravity.

💡Potential Flow

Potential flow is a type of flow where the velocity field is the gradient of a scalar potential. In the video, potential flow is mentioned in the context of irrotational flows, with the script explaining that the curl of a gradient is always zero, indicating that potential flows are inherently irrotational.

💡Conservation Laws

Conservation laws are principles that state certain quantities remain constant in a closed system. In the video, conservation laws are mentioned as a future topic, with the script suggesting that the understanding of divergence, gradient, and curl will be crucial in deriving partial differential equations related to these laws.

Highlights

Introduction to vector calculus operations, focusing on the divergence, gradient, and curl.

Explanation of the curl as a measure of rotation in a vector field.

Analysis of two vector fields to illustrate the concept of divergence and curl.

Definition of the curl as the cross product of the del operator and a vector-valued function.

Description of the del operator in three dimensions for calculating the curl.

Computational method for the curl using the determinant of a 3x3 matrix.

Example calculation of the curl for a specific vector field.

Interpretation of the curl in terms of the rotation of a vector field.

Introduction to the concept of solid body rotation and its relation to the curl.

Application of the curl in fluid flows and mechanical objects like asteroids.

Computation of the curl for 2D vector fields and its interpretation.

Explanation of curl-free or irrotational vector fields and their properties.

Analysis of a vector field with both divergence and rotation.

Demonstration of how to compute the curl for a vector field representing solid body rotation.

Derivation of the curl of a vector field from the angular velocity vector.

Important properties of the curl: the curl of a gradient is always zero.

Important properties of the curl: the divergence of a curl is always zero.

Conclusion on the significance of the curl in understanding vector fields and preparing for conservation laws and partial differential equations.

Transcripts

[Music]

all right welcome back

so we're talking about the divergence

the gradient and the curl of

vector fields these are vector calculus

operations and in the last two lectures

i covered the grad and div so today

we're going to talk about the curl

of our vector field

and curl is a really useful quantity it

tells us how much our vector field is

rotating so these were the two vector

fields i looked at in the last lecture

for divergence where this one is

is diverging and this one has divergence

zero

but we noticed if we just look at this

one by eye

this is kind of symmetrically

outsourcing in the radial direction

there's no rotation to this vector field

so what we hope is that this curl

quantity will be zero for this case

and in this vector field which has

divergence zero we notice that our flow

essentially is just rotating objects and

so we would expect that we would have a

non-zero curl for this case

so i'm going to define the curl show you

how to compute it we're going to look at

it for these two examples and we'll

think about some other kind of

applications of the curl in fluid flows

and in things like solid body rotations

of like asteroids and you know

mechanical objects

okay good so the curl of a vector field

is essentially this differential

operator del cross product

and

remember i'm just going to write down

the del operator so del or nabla

whatever this differential operator here

and since we're doing a cross product

i'm actually going to write this in

three

dimensions before i've been usually

doing this in 2d just for simplicity

here i'm going to write this in 3d as

partial partial x

partial partial y and partial partial z

or z

okay that's my dell operator

and we're going to take the cross

product of del with a vector valued

function f so f

bar

is going to be a function that has three

components f1

f2

and f3

and just to be very explicit because

this is a vector-valued function of

three components it's a component of x y

and z these are functions of x y and z x

y and z or z

x y and z

x y and

z

okay so that's my vector valued function

and just like these are two dimensional

vector fields this would be a

three-dimensional vector field it would

describe how vectors are pointing in all

of three space

and this could be a nonlinear function

this f1 f2 and f3 could be non-linear

functions of x y and z so for example i

could have a vector field that does

something here and does something

totally different over here based on

this nonlinear function

and so the curl of f uh the way we write

this is sometimes the curl of f

so

curl of f

is equal to the cross product

of my del operator with my f vector okay

so cross product of this vector with

this vector

and so we would literally do uh the

normal cross product that we're used to

seeing so this is equal to

uh the determinant so i'm just going to

write down like the determinant

of this 3x3 matrix that has an i a j

and a k component so this is how we do

the cross product just remember from

from high school we take the determinant

of the ijk unit vectors the second row

is my first vector which in this case is

del so

partial partial x

partial partial y and partial partial z

or z

and the third component of this curl of

this determinant matrix we're writing

down is my f function here f 1

f 2

and f 3.

and so you can actually compute this and

write this out

if you have these f components you can

compute the determinant of this matrix

and it is a expression that will have an

i a j and a k component

so this is really important

the curl takes in a vector field f f bar

a multi-component vector field

and it spits out another vector field it

will have an i a j and a k component so

it takes in a vector and it outputs a

vector

now that's different than the div and

grad operators so grad takes in a scalar

and returns a vector

div takes in a vector and returns a

scalar

and curl takes in a vector and returns a

vector

and so this is kind of interesting

actually if you think about why these

are the three kind of primitive linear

differential operators that we use for

all of linear partial differential

equations and vector calculus

i think there is some intuitive kind of

reason why we need these three operators

to kind of take me from scalar to vector

from vector to scalar and from vector to

vector okay and i guess there's a fourth

one which is the laplacian operator del

squared which takes me from scalar to

scalar okay so i can kind of go from any

of those quantities

let's do this on an example let's say

that my function f f bar equals

uh i think i have an example here that i

like let's say it equals

x y

minus sine

z or z and one okay so this is my f1 f2

and f3 components

and if you like you can also write this

as you know x y

in the i direction

minus sine z in the j direction plus 1

in the k direction so i want you to be

very comfortable going back and forth

between this i j k coordinate notation

and this kind of column vector notation

these are our equivalent representations

this is usually what physicists use this

is usually kind of what i use applied

math

differential equations notation here but

they're completely equivalent

but because i'm using this notation for

the curl for the determinant of this

this thing i want to show you uh how

simple it is to go back and forth from i

j k to column vectors

okay so we're literally going to compute

this quantity the curl of f for this

very specific f vector here so i'm

literally just going to

swap out

f1 f2 and f3 for these quantities here

which is x y

minus sine

z

and one

okay good

uh and

now we actually just go through the the

kind of algebra of computing this this

determinant so i'm just going to remind

you i'm going to do it kind of quickly

the way we do it is

to compute this determinant i basically

take uh the i component

times the determinant of this 2 by 2

subblock that's kind of opposite it so

that's partial y

partial partial y of 1

minus partial partial z of minus sine z

so in the i component

i have this little subdeterminant which

is zero

minus

minus cosine of z so plus cosine of z

i did that a little quickly but you can

confirm that this determinant is just

cosine of z

then i have okay so that was the first

block plus i direction then i do minus

the j direction so these we always

alternate signs as we go along these uh

these kind of majors so minus in the j

direction

the determinant of the two by two sub

block excluding this column

so that would be partial partial x

of one which is

zero minus partial partial z of x y

which is also zero

so i have zero in the j uh the j

component so there's nothing in the j

direction of this curl

and then again so plus i minus j plus k

we alternate sines plus

in the k direction we have the

determinant of this two by two sub block

so partial partial x of sine z again

that's zero

minus partial partial y of x y is just

minus x

minus

x

and so the

curl of this vector field the curl of

this vector field

is a function

that is uh cos

z in the i direction

minus x in the k direction

and if you like you can write that as a

vector

cos z

0

minus x i'm sorry my pen is squeaky i

know that that drives everyone crazy

uh so really really simple this is how

we compute the curl of any old vector

field you like you can write down any

vector field you want and it's

relatively simple to compute its curl

okay now the interpretation of the curl

is a little more subtle what does this

mean what does it mean to have a curl

that's cos z 0 minus x and i'll talk

about that right now

okay

so

let's let's now compute the curl on

these simpler 2d vector fields okay and

so the way we're going to think about

these 2d vector fields

is essentially we have

f1 f2 and 0.

and so if you have f1 f2 and zero

then all of our curl is going to be uh

in the

k

direction okay and so i'm just going to

kind of like write out what this what

this would be in these cases okay

so let me do this in pink here so if i

want to compute the curl of a 2d uh

f bar like this

i would say that in the eye direction

it is

uh oh my goodness let me think um

yeah i'm sorry it's always in the k

direction

and it's going to be partial

f2

partial f2

partial x

minus partial f1 partial y

minus partial f1

partial y this is the curl in 2d this is

just how you and this is all

uh in the unit normal k direction kind

of going out of the board the z

direction that we haven't plotted here

this is x this is y we assume that this

curl is kind of perpendicular to the

board to be consistent with with this

notation but this is the quantity we

compute so let's actually compute it

here okay so the curl of this vector

field curl

of this particular vector field is

partial

f2 partial x that's partial y partial x

that is zero

minus partial f1 partial y which partial

x partial y is also zero

and so this vector field uh is curl free

so we say that this is curl

free

or

ear irrotational arrow oh my goodness

tatiana probably two t's irrotational oh

my goodness i can't spell

i can't spell at all

irrotational so if the curl of a vector

field is zero

it is curl free or irrotational i think

that's probably uh spelled wrong

and this is an intuitive case okay so

this vector field is just you know kind

of radially sourcing out it's symmetric

and if i have a particle that starts at

some angle it just blasts off in that

same angle forever it never kind of

rotates around in this phase portrait

and so that's kind of the intuition of

what it means to be curl free

if there is no rotational component of

this vector field the vector field is

said to be curl free or irrotational

okay

uh good

and of course there's not two t's in

here um because if i had just a regular

rotational vector field rotate is

spelled with one t rotational good

uh okay

but this vector field down here we have

some reason to think that this is

rotational because again if i start with

a little particle or blob here it

rotates around this vector field we've

kind of seen uh seen that happen

so let's compute the curl of this guy

here

okay so similarly

curl of f bar

is

partial of the second component with

respect to x

minus partial of the first component

with respect to y

so partial of the second component with

respect to x is one

minus

partial of the first component with

respect to y is minus minus one which is

plus one

equals two

so this vector field has a curl of two

or it is it is rotational so this is

a rotational

it is a rotational vector field

and i picked a really easy case where

this vector field was divergence free

and had rotation

this vector field

was rotation free but had a divergence

you can have vector fields that have a

little bit of both that are kind of

swirling and expanding that would be

like an unstable spiral source or i

could have a

vector field that's spiraling in

and so that would be a negative

divergence uh

negative curl kind of situation

okay so this hopefully gives you an idea

of the intuition of what this curl means

intuitively on these vector fields

what else do i want to tell you

okay so this had a

positive curl

because this follows the right hand rule

this vector field is curling with my

right hand where my thumb is sticking in

the positive out of board direction this

is just the convention we've agreed on

this is a positive two curl

if i had a vector field that was going

the other direction

that literally i just said this was um i

guess like

minus x and plus y

then this would be going the other way

and my thumb would be going into the

board that would be a negative 2 that

would be rotating in the negative

direction using the right hand rule so

this is just a convention that we all

agree on i think it's a bit prejudiced

i'm left-handed but i'm forced to live

in a right-handed

uh curl world that's okay just get used

to the the convention here

okay um good so we know how to compute

this thing for generic arbitrary nasty

three component vector fields and we get

our own

curl vector field out so that's kind of

interesting we also know how to compute

it for 2d

vector fields you basically just look at

the kth component and that little two by

two sub block

uh here

now i want to give you a little more

intuition for how you can use this curl

so um

maybe i'll just do it right here

and i'll show you so how we can

interpret curl

for something like a solid body rotation

so this i think of as a fluid flow okay

this vector field is like

the vector field that you would see if

you looked at the gulf of mexico or if

you turned on your faucet in your sink

or in your bathtub you might get uh you

know a vector field on the the bottom

metal pan everything's spreading out

and so you know these are vector fields

that are very intuitive from a fluids

perspective

but what i want to show you is that you

can also think about this for things

like if i have an asteroid that's

rotating in space that also has an

interpretation with the curl okay so

let me just

finish this and show you how that that

situation looks

okay

uh so how do i want to draw this i am

going to draw

a

asteroid or some just object this is

just some three-dimensional object

and it lives in x y z coordinates so

there's some coordinate system

uh x

y

and z or z

and again we're using the right hand

rule notation so that if i curl from x

to y i my thumb sticks up in the z

direction

uh good

and what we're going to do is assume

that this asteroid is spinning about

some axis and i'm going to make it

really easy and say that it's spinning

about the z or z axis

so we're going to define this

w vector to be the vector around which

this object is spinning

and we're going to say that it is

rotating at a rate given by omega and i

know that my w and my omega look

identical

but this little arrow tells you that

this is a vector and this doesn't have

an arrow it's a scalar this is the

angular rate

rotating about this vector w

and in general this vector w could be

pointing any which way you like i'm just

picking it aligned with z because it's

easier for me okay

good

now a physical intuitive interpretation

of curl

is that if i have some point on this

asteroid or inside the asteroid let's

say i have some point

i'm going to call it q

and literally from the origin to q is a

vector so every point in this asteroid

has some vector q from the origin to

that point

then that point is moving at some

velocity because the whole thing is

rotating so this is moving at some

velocity that's kind of you know

perpendicular it's kind of moving in the

direction

tangent to this rotation around this w

axis

so this has some v vector

of motion this cue point

has some v

motion that's induced by this whole

object rotating okay and i'm going to

say that this point q is given by a

radial vector r from the origin so q is

defined as some r vector from the origin

and it is rotating this the the rate of

change of this is another vector uh

v v

and so the physical interpretation of

curl that you've probably all learned in

your physics class or in your calculus

class

is that

this vector v

is equal to my omega vector

cross product with my r vector

so every single point on this asteroid i

define that r vector from the origin to

that point and to figure out what the

velocity of that point is i just take

the curl of my the vector i'm rotating

about

with that vector that defines that

position this is the velocity of every

position r that's rotating about that

axis w

so this is pretty cool this is a useful

interpretation of the curl

where essentially this is the angular

velocity vector so i'm just actually

going to write that so this is the

angular

velocity

vector

okay good uh and so if we have

so now uh let me write if

assume that this this w vector if uh

this w vector is just omega

in the k direction in the z

direction kind of in the z third

component if omega is just rotating at

some constant rate omega in the z

direction

and if we say uh and

r equals

x in the i direction plus y in the j

direction plus z in the k direction

that's just how we specify the x y and z

coordinates of this point q x in the i

direction

i'll take x steps in the i direction

y steps in the j direction and z steps

in the the k direction that defines q

then i can compute this v

by taking the cross product of these two

things

um good and so v

equals the curl of these two objects um

and if i want to do that i'm just going

to leave this as an exercise for you all

you can compute this yourself

you get essentially w cross r

equals

minus omega y

in the i direction

plus omega

x

in the j direction

and this makes sense because if i have

some point that is aligned with this

axis in the in the k direction

that point's actually not moving it has

a zero velocity because it's just kind

of spinning around

its axis it's not actually moving you

know with a with a

non-zero velocity so it makes sense that

this curl if i'm rotating about the z

axis would only have components in the i

and j

coordinates

but this is where it gets really

interesting is that if we take the curl

so this v

object this v is a vector field it is

literally a vector field every point on

this asteroid

now defines a vector of how fast that

point on the asteroid is moving around

as this thing spins

so for every point you know points that

are closer to the axis of rotation are

moving less fast points that are farther

from the axis of rotation are moving

more fast points that are at the very

tips are moving the fastest and that

establishes this vector field

for how this thing is rotating

now you could actually

convert this to radial coordinates to

cylindrical coordinates and you would

find that this is uh does not depend on

theta but it only depends on the radius

from that that axis of rotation so again

something you should do

as an exercise

but it gets really interesting so if i

take the curl of this vector field

so curl of

a v

is equal to i have the i

vector the j vector and the k vector

and i have partial partial x partial

partial y

partial partial z

and now i have you know minus omega y

plus omega x and zero

because that's literally what

you're going to derive as a homework

problem is that this is the the v vector

then the curl of that we can compute

pretty easily

okay so in the i direction partial

partial y of zero is zero partial

partial z of x y is zero

so this is zero in the i direction

similarly in the j direction partial

partial x of zero is

partial partial z of omega y is 0.

so

plus 0 in the j direction

and now in the k direction i have

partial partial x of omega x that's an

omega

minus partial partial y of minus omega y

is another

plus omega

so i get plus 2 omega

in the k direction

and so what this means is that if i have

a solid body rotation

literally the rotation of a solid body

like an asteroid

of angular rate omega

about some axis uh you know like the k

direction or the z axis

the curl of the vector field of all of

those particles is equal to two times

that angular rate about the axis of

rotation

and so if i had been rotating about a

different axis

this would be uh in that direction of

that other other axis okay

so that's a really powerful intuitive

definition of curl is that a positive

constant curl

this is just a constant this is not a

function of x y and z it's a constant in

the k direction

a constant positive curl

is associated with a solid body rotation

literally like all the geometry of this

body doesn't deform or change as it's

rotating

and similarly here

because i had a constant uh

curl in this vector field this also

corresponds to solid solid body rotation

if i have some little blob or some

object in this flow it won't get

deformed as it's moving around it will

keep its shape and just rotate as if it

was a solid body in this fluid flow

so very very useful concept here of

solid body rotation

okay i realize that i'm uh just

massively running out of time here and i

don't want to overload you but a couple

of last things i'll tell you

really important facts

the curl

of

a gradient

always equals zero

so the curl

of the gradient of any f

equals zero for

all

f

very interesting facts so any these are

uh are called potential flow solutions

these potential flow vector fields any

potential flow vector field is

irrotational so the gravitational uh

vector field on earth is a potential

field this is literally the gradient of

the gravitational potential that is an

irrotational vector field things are

just coming in at a constant rate

and if you if you come in at some angle

you'll stay on that approach vector

so the curl of a gradient is always zero

and similarly the divergence of a curl

is always zero

of curl

equals 0.

so the divergence of the curl

of any vector-valued function f

equals 0

again for

all

f

and this is kind of cool if i have a

vector field and i compute its curl

it actually the curl of a vector field

gives me exactly the components of that

vector field that are rotating so if

this vector field had a divergence

component and a curl component if it was

you know swirling

and expanding

the curl would only pull out the

swirling part of that vector field it

would only pull out the rotational part

and it would always have a divergence

free

vector field the curl of any vector

field is always divergence free

so we'll we'll revisit these later maybe

we'll even derive these but i just

wanted to throw these up here very very

important properties of the curl as they

relate to the gradient and the

divergence of a vector field

okay good so now we've covered the

gradient the divergence and the curl

we're ready to start looking at how we

would write down conservation laws

uh you know in a control volume and

starting to derive partial differential

equations

for real physical systems so that's all

coming up soon uh thank you