Gradient Diver Curl

Gradient, Divergence & Curl

Chapter 1

2

Tc

Td

Scalar and vector fields

va

vb

Fluid

Imagine a cooling system of a reactor which is using fluid as the cooler medium

3

FIELD is a description of how a physical quantity varies from one point to another in the region of the field (and with time).

(a) Scalar fields

Ex: Depth of a lake, d(x, y)

Temperature in a room, T(x, y, z)

Depicted graphically by constant magnitude contours or surfaces.

y

x

d1

d2d3

4

At any point P, we can measure the temperature T. The temperature will depend upon whereabouts in the

reactor we take the measurement. Of course, the temperature will be higher close to the radiator than the opening valve.

Clearly the temperature T is a function of the position of the point. If we label the point by its Cartesian coordinates , then T will be a function of x, y and z, i.e.

. This is an example of a scalar field since temperature is

a scalar. ),,( zyx

),,( zyxTT

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1. A field is a quantity which can be specified everywhere in space as a function of position.

2. The quantity that is specified may be a scalar or a vector.

3. For instance, we can specify the temperature at every point in a room.

4. The room may, therefore, be said to be a region of “temperature field” which is a scalar field because the temperature T (x, y, z) is a scalar function of the position.

5. An example of a scalar field in electromagnetism is the electric potential.

6

Meanwhile, at each point, the fluid will be moving with a certain speed in a certain direction

That is, each small fluid element has a particular velocity and direction, depending upon whereabouts in the fluid it is.

This is an example of a vector field since velocity is a vector. The velocity can be expressed as a vector function, i.e.

where will each be scalar functions.

kjivv ),,(),,(),,(),,( 321 zyxvzyxvzyxvzyx

321 , vvv and

7

1-7Example: Linear velocity vector field of points on a rotating disk

8

Physical examples of scalar fields:

Electric potential around a charge

Temperature near a heated wall

+

(The darker region representing higher values )

9The flow field around an airplane

Electric field surrounding a positive and a negative charge.

Hurricane

+ −

Magnetic field lines shown by iron filings

Physical examples of vector fields:

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Gradient of Scalar Fields

The gradient of a scalar field is a vector field,

which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

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grad

x, y, z ctt

dr��������������

Physical meaning: is the local variation of Φ along dr. Particularly, grad Φ is perpendicular to the line Φ = ctt.

grad dr��������������

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Gradient operator

Suppose , we have a function of three variables- say, the temperature T(x, y, z) in a room.

For the temperature distribution we see how a scalar would vary as we moved off in an arbitrary direction.

Now a derivative is supposed to tell us how fast the function varies, if we move a little distance.

13

kz

Tj

y

Ti

x

TTgrad

z

ky

jx

i TTgrad

If T(r) is a scalar field, its gradient is defined in Cartesian coordinates by

It is usual to define the vector operator

,

which is called “del”. We can write

.

Without thinking too hard, notice that grad T tends to point in the direction of greatest change of the scalar field T.

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dzz

Tdy

y

Tdx

x

TdT

ldT

kdzjdyidxkz

Tj

y

Ti

x

TdT

The significance of the gradient:

A theorem on partial derivatives states that

This rule tells us how T changes when we alter all three variables by the infinitesimal amounts dx, dy, dz. Change in T can be written as,

The conclusion is that, the RHS of above equation is the small change in temperature T when we move by dl.

15

.

ldofdirectiontheinvectorunitaisdl

ldbut

dl

ldT

dl

dT

getWe

^^

aTaofdirectionindl

dT

If we divide the above eq. by dl

So , we can conclude that, grad T has the property that the rate of change of T w.r.t. distance in any direction â is the projection of grad T onto that direction â.That is

In general,

• a directional derivative had a different value for each direction,

• has no meaning untill you specify the direction

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dl

dT is called a directional derivative.

the quantity

Gradient Perpendicular to T constant surfaces

If we move a tiny amount within the surface, that is in any tangential direction, there is no change in T , so

  Surface of constant T,

These are called level surfaces. Surfaces of constant T

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.0dl

dT

0

dl

ldTsurfacethein

dl

dl

Conclusion is that; grad T is normal to a surface of constant T.

So for any

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Geometrical Interpretation of the Gradient

Like any vector, a gradient has magnitude and direction. To determine its geometrical meaning, lets rewrite the dot product In its abstract form:

cosTdldlTdT

(1.5)

where θ is the angle between and dl. Now, if we fix the magnitude dl and search around in various directions (that is, vary θ), the maximum change in T evidently occurs when θ =0 (for then cos θ = 1). That is for a fixed distance dl, dT is greatest when I move in the same direction as .

T

T

19

In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

20

Example 1

If (x,y,z) = 3x2y– y2z2, find grad and at the point (1,2,−1).

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Solution

kji

kji

)2()23(6

grad

222 zyyzxxy

zyx

At the point (1,2,−1),

kji

kji

812

)1()2(2])1)(2(2)1(3[)2)(1(6 222

2098)1(12812 222

)1,2,1(

kji

Example 2

If

2

^

2

^^^

222

1)(

2)()(

.

?

r

r

rb

rra

thatShowQ

kzjyixrhere

routFind

zyxr

23

Example 3

Find , if

Given

ji )(sin cos yexxy

),( yx

.0)0,0(

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SolutionSince , we haveji )(sin cos yexxy

.....(1) cos xyx

.....(2) sin yex

y

Integrating (1) and (2) w.r.t. x and y respectively, we obtain

.....(3) )(sincos yfxyxdxy

.....(4) )(sin)(sin xgexydyex yy

25♣

Comparing (3) and (4), we can conclude that

Ceyf y )(

Hence, Cexyyx y sin),(

To find constant C, use .0)0,0(

1

01

00sin0)0,0( 0

C

C

Ce

Therefore, 1sin),( yexyyx

and Cxg )(

where C is an arbitrary constant of integration

26

Example 4

Find if

and .

),,( zyx

kji )34()23()2( 2233232 yzxzzxxyxyzy 2)0,0,0(

27

Integrating (1), (2) and (3) w.r.t. x, y and z respectively, we obtain

SolutionWe have .....(1) 2 32 xyzy

x

.....(2) 23 32zxxyy

.....(4) ),()2( 32232 zyfyzxxydxxyzy

.....(3) 34 223 yzxzz

.....(5) ),(3)23( 32232 zxgyzxxyydyzxxy

.....(6) ),()34( 324223 yxhyzxzdzyzxz

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Comparing (4) with (5) and (6) we get

Czyzyf 43),(

Czyyzxxy 4322 3

To find constant C, use 2)0,0,0(

23 4322 zyyzxxy

Therefore

Problem 5

29

0.

),,(

.),,(

^

rd

kdzjdyidxz

ky

jx

i

odzz

dyy

dxx

d

kzyx

constiskwherekzyxsurfacetheto

larperpendicuvectoraisthatShow

30

Application of gradient: Surface normal vector

A normal, n to a flat surface is a vector which is perpendicular to that surface.

A normal, n to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P.

n

),( yxfz

31

Therefore, for a non-flat surface, the normal vector is different, depending at the point P where the normal vector is located.

Unit vector normal is defined as

n

),( yxfz

n

nn ˆ

32

To find the unit vector normal to the surface

we follow the following steps:

(i) Rewrite the function as

(ii) Find the normal vector that is

(iii) Then, the unit normal vector is

(iv) Hence, the unit normal vector at a point

is

n

nn̂

),( yxfz ),( yxfz

constantany ,),,( kkzyx n

),,(

),,(ˆ

000

000

zyx

zyx

n),,( 000 zyxP

33

Example 5

Find the unit normal vector of the surface at the indicated point.

(a) at

(b) at

226 yxz 23 zyxe y

)2,3,1(

)1,0,1(

34

Solution

(a) Rewrite as2214 yxz 14222 zyx

Thus, we obtain 222),,( zyxzyx

Then, )(2222 kjikji zyxzyx

At the point (−1,3,2),

The unit normal vector is14

23ˆ

kjin

14223)1(2 222 and)23(2 kji

35

(b) Rewrite as

Thus, we obtain

Then,

At the point (1,0,−1),

The unit normal vector is6

2ˆ

kjin

6211 222 and

23),,( zyxezyx y

kji zyxee yy 2)3( 2

23 zyxe y 023 zyxe y

kjikji 2)1(2])0(31[ 200 ee

36

Divergence of Vector Fields

The divergence is an operator that measures the magnitude of a vector field's source or sink at a given point

The divergence of a vector field is a scalar

(x, y, z)V��������������

(x dx, y, z)V��������������

x x+dx

37

The divergence of a vector field

is defined as

z

F

y

F

x

F

FFFzyx

321

321 )(

div

kjikji

FF

kjiF ),,(),,(),,(),,( 321 zyxFzyxFzyxFzyx

38

Divergence

y ux zu

vv vdiv = v

x y z

v v����������������������������

2 – Physical meaning

(x, y, z)v��������������

is a differentiable vector field

div v��������������

is associated to local conservation laws: for example, we will show how that if the mass of fluid (or of charge) outcoming from a domain is equal to the mass entering, then is the fluid velocity (or the current) vectorfield

div 0v��������������

v��������������

(x, y, z)V��������������

(x dx, y, z)V��������������

x x+dx

Geometrical Interpretation.

The name divergence is well chosen, for . F is a measure of how much the vector F spreads out (diverges) from the point in question.

The vector function has a large (positive) divergence at the point P; it is spreading out. (If the arrows pointed in, it would be a large negative divergence.)

P

39

NOTE: P=electric field due to charge (+ ve or – ve)

On the other hand, the function has zero divergence at P; it is not spreading out at all.

40

P

So, for example, if the divergence is positive at a point, it means that, overall, that the tendency is for fluid to move away from that point (expansion); if the divergence is negative, then the fluid is tending to move towards that point (compression).

Fundamental theorem of divergence

The fundamental theorem for divergences states that:

41

surfacevolume

daFdF

This theorem has at least three special names: Gauss’s theorem, Green’s theorem, or, simply, the divergence theorem.

is function at the boundary element of volume (in Cartesian coordinates, = dx, dy, dz), and The volume integration is really a triple integral.

d

d

d

da represents an infinitesimal element of area; it is a vector , whose magnitude is the area of the element and whose direction is perpendicular ( normal ) to the surfaces, pointing outward.

42

On the front face of the cube, a surface element is idzdyda ˆ

1

43

jdxdzda ˆ2

on the right face, it would be

whereas for the bottom it is

kdydxda ˆ3

Problem 1

Q. Calculate the divergence of the following vector functions?

zxkyzjxyivb

xzkjxzixva

32)(

23)(

2

221

Problem 2

Check the divergence theorem using the function

45

kyzjzxyiyv )2()2( 22

And the unit cube is situated at the origin.

Z

Y

X

46

Curl of Vector Fields

Curl is a vector operator that shows a vector field's rate of rotation, i.e. the direction of the axis of rotation and the magnitude of the rotation.

0curl v��������������

0 v

47

48

The curl of a vector field

is defined as

kjiF ),,(),,(),,(),,( 321 zyxFzyxFzyxFzyx

y

F

x

Fk

z

F

x

Fj

z

F

y

Fi

FFFzyx

121323

321

curl

kji

FF

Problem 1

Find the curl of

0

3

xyzyx

kji

v

xjyiv 3

The curl of v3 points in the z-direction

Z

y

X

Curl          To find a possible interpretation of the curl, let us consider a body rotating with uniform angular speed about and axis l. Let us define the vector angular velocity      to be a vector of length extending along l in the direction Take the point O as the origin of coordinates we can write R = xi + yj + zk                                                     

the radius at which P rotates is |R||sin| Hence, the linear speed if P isv = |R||sin| = |R||sin|

If we take the curl of V, we therefore have

that is

Expanding this, remembering that Ω is a vector, we find

Conclusion: The angular velocity of a uniform rotating body is thus equal to one-half the curl of the linear velocity of any point of the body.

^^

cossin jyzexiyzexu xx

wvuzyx

kji

u

^^^

^^

cossin jyzexiyzexu xx

^^^

^^^

^^^

coscos1cossin

sincossincos

0cossin

kyzzeyzejyzyeiyzye

kyzexy

yzexx

jyzexz

iyzexz

yzexyzexzyx

kji

u

xxxx

xxxx

xx

Example: For velocity field,

, find the angular velocity ω.

For the field,

, we obtain:

Fundamental theorem of curl

The fundamental theorem for curls, which goes by the special name of Stokes’ theorem, states that

lineboundarysurface

dlvdav .

The integral of a curl over a region (a patch of surface) is equal to the value of the function at the boundary (the perimeter of the patch).

54

55

Example 6

Find both div F and curl F at the point (2,0,3) if

kjiF )2(cos2),,( 2 yxyxzzezyx xy

56

Solution

yxzyze

yxz

yxzy

zex

z

F

y

F

x

F

xy

xy

sin22

)2()cos2()(

div

2

2

321

FF

At the point (2,0,3),

00sin)3)(2(2)3)(0(2 )0)(2(2 eF

[Notice that div F is a scalar!]

57

kji

k

j

i

kji

FF

)2cos2()1()cos22(

)()cos2(

)()2(

)cos2()2(

2cos2

curl

22

2

2

2

xyxy

xy

xy

xy

xzeyzeyx

zey

yxzx

zez

yxx

yxzz

yxy

yxyxzzezyx

58

At the point (2,0,3),

ki

k

jiF

62

])3)(2(20cos)3(2[

]1[]0cos)2(22[)0)(2(2

)3)(0(2

e

e

[Notice that curl F is also a vector]

59

Properties of Del

If F(x,y,z) and G(x,y,z) are differentiable vector functions (x,y,z) and (x,y,z) are differentiable scalar functions, then

(i)

(ii)

(iii)

)(

)(

2

60

(iv)

(v)

(vi)

(vii)

(viii)

GFGF )(

00 grad curlor )(

GFGF )(

2

2

2

2

2

2

2

2

2

2

2

22

)(

zyx

zyx

0 curl divor 0)( FF

*Notes: In (vi), is called the Laplacian operator2