Dr. Hugh Blanton ENTC 3331 Gradient, Divergence and Curl: the Basics

Dr. Hugh Blanton

ENTC 3331

Gradient, Divergence and Curl: the Basics

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 3

• We first consider the position vector, l:

• where x, y, and z are rectangular unit vectors.

zyxl ˆˆˆ zyx


• Since the unit vectors for rectangular coordinates are constants, we have for dl:

dzdydxd zyxl ˆˆˆ


• The operator, del: is defined to be (in rectangular coordinates) as:

• This operator operates as a vector.

zyx

zyx ˆˆˆ


Gradient Gradient

• If the del operator, operates on a scalar function, f(x,y,z), we get the gradient:

z

f

y

f

x

ff

zyx ˆˆˆ


• We can interpret this gradient as a vector with the magnitude and direction of the maximum change of the function in space. • We can relate the gradient to the

differential change in the function:

dzz

fdy

y

fdx

x

fdf


Directional derivatives:

aTdl

dTlˆ

lal

ddl ˆ

ldfdzz

fdy

y

fdx

x

fdf


• Since the del operator should be treated as a vector, there are two ways for a vector to multiply another vector: • dot product and • cross product.


Divergence Divergence

• We first consider the dot product:• The divergence of a vector is defined to be:

• This will not necessarily be true for other unit vectors in other coordinate systems.

z

A

y

A

x

A

AAAdzz

dyy

dxx

zyx

zyx

zyxA ˆˆˆ


• To get some idea of what the divergence of a vector is, we consider Gauss' theorem (sometimes called the divergence theorem).


Gauss' Theorem (Gau’s Theorem Gauss' Theorem (Gau’s Theorem

• We start with:

dydzdxz

A

y

A

x

AdV zyx

A

dxdydzz

Adxdzdy

y

Adydzdx

x

AdV zyxA

Surface Areas


• We can see that each term as written in the last expression gives the value of the change in vector A that cuts perpendicular through the surface.


• For instance, consider the first term:

• The first part:

• gives the change in the x-component of A

dydzdxx

Ax

dxx

Ax


• The second part,

• gives the yz surface (or x component of the surface, Sx) where we define the direction of the surface vector as that direction that is perpendicular to its surface.

dydz


• The other two terms give the change in the component of A that is perpendicular to the xz (Sy) and xy (Sz) surfaces.


• We thus can write:

• where the vector S is the surface area vector.

S S

ddddV SASASAA


• Thus we see that the volume integral of the divergence of vector A is equal to the net amount of A that cuts through (or diverges from) the closed surface that surrounds the volume over which the volume integral is taken. • Hence the name divergence for

A


• So what?• Divergence literally means to get farther

apart from a line of path, or• To turn or branch away from.


• Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles:

Goes straight ahead at constant velocity.

(degree of) divergence 0


Now suppose they turn with a constant velocity

diverges from original direction

(degree of) divergence 0


Now suppose they turn and speed up.

diverges from original direction

(degree of) divergence >> 0


Current of water

No divergence from original direction

(degree of) divergence = 0


Current of water

Divergence from original direction

(degree of) divergence ≠ 0


E-field between two plates of a capacitor.

Divergenceless

+


-field inside a solenoid is homogeneous and divergenceless.

I

divergenceless solenoidal


CURL


• Two types of vector fields exists:

+

+E

Electrostatic Field where the field lines are open and there is circulation of the field flux.

β

Magnetic Field where the field lines are closed and there is circulation of the field flux.

circulation (rotation) = 0 circulation (rotation) 0


• The mathematical concept of circulation involves the curl operator.• The curl acts on a vector and generates

a vector.

zyx BBBzyx

zyx

B

ˆˆˆ


• In Cartesian coordinate system:

zyx BBBzyx

zyx

B

ˆˆˆ


• Example

zyx BBBzyx

zyx

B

ˆˆˆ


• Important identities:

0 V for any scalar function V.

BABA

0 A


Stoke’s TheoremStoke’s Theorem

• General mathematical theorem of Vector Analysis:

Css

ddsd lBBsB

Any surfaceClosed

boundary of that surface.


• Given a vector field

• Verify Stoke’s theorem for a segment of a cylindrical surface defined by:

r/cosˆ zB

30

,23

,2

z

r


y

x

z

rn ˆˆ

Cs

dd lBsB


zr BrBBzr

r

r

zφr

B

ˆˆˆ1


zr BrBBzr

r

r

zφr

B

ˆˆˆ1

z

BB

rz

1

r̂


zr BrBBzr

r

r

zφr

B

ˆˆˆ1

z

B

r

B

z

BB

rrzz φr ˆ

1ˆ


zr BrBBzr

r

r

zφr

B

ˆˆˆ1

rrzz B

rr

Bz

z

B

r

Bφ

z

BB

rr

1ˆˆ

1ˆ


• Note that has only one component:

rrzz B

rr

Bz

z

B

r

Bφ

z

BB

rr

1ˆˆ

1ˆ

rBz /cos

B

22

cosˆ

sinˆ

cos

ˆ

cos1ˆ

rrrrr

r

φrφr


3

0

2

322

ˆcosˆ

sinˆ

zdzrd

rr

rφr

sB

ds

The integral of over the specified surface S isB


4

303

4

1

4

1

2

10

2

1cos

2

1

cos2

1sin

3

0

3

0

3

0

2

3

3

0

2

3

3

0

2

3

zzz

zz

dzdzdz

dzdzdr


y

x

z

rn ˆˆ

Cs

dd lBsBa

b

c

d


C

d lB

The surface S is bounded by contour C = abcd.

The direction of C is chosen so that it is compatible with the surface normal by the right hand rule.r̂

a

d da

d

c cd

c

b bc

b

a ab

C

dBdB

dBdBd

ll

lllB


a

d

d

c

c

b

b

a

dzzr

zrdφr

z

dzzr

zrdφr

z

ˆcosˆˆ

cosˆ

ˆcosˆˆ

cosˆ

0

3

3

2

3

0

2

3

ˆcosˆˆ

cosˆ

ˆcosˆˆ

cosˆ

dzzr

zrdφr

z

dzzr

zrdφr

z


0

3

3

2

3

0

2

3

ˆcosˆˆ

cosˆ

ˆcosˆˆ

cosˆ

dzzr

zrdφr

z

dzzr

zrdφr

z

0

3

3

0ˆ

cosˆˆ

cosˆ dzz

rzdzz

rz

0

3

3

0

0

3

3

0 23

cos

22

cos

23

cos

22

cosdzdzdzdz


4

3

3023

cos

23

cos

23

cos0

3

0

3

zdz


CurlCurl


dzz

fdy

y

fdx

x

fdf

dzz

fdy

y

fdx

x

fdf


curl or rot

• place paddle wheel in a river• no rotation at

the center• rotation at the

edges


• the vector un is out of the screen• right hand rule s is surface

enclosed within loop• closed line integral


Electric Field LinesElectric Field Lines

Rules for Field Lines

1. Electric field lines point to negative charges

2. Electric field lines extend away from positive charges

3. Equipotential (same voltage) lines are perpendicular to a line tangent of the electric field lines

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Dr. Hugh Blanton ENTC 3331 Gradient, Divergence and Curl: the Basics