probability and stochastic processes 2nd roy d yates and david j goodman

Lecture 7 Curl and Laplacian Operators

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Mathematic Operators in EM study

• Gradient

• Divergence

• Curl

• Laplacian

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Highlights

• Concept of Circulation

• Derivation of x B

• Stoke’s theorem

• Definition of 2V

• Definition of 2E

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What is Curl? • The curl of a vector is a measure of the circulation of

the vector field per unit area s, with the orientation of the unit area s chosen such that the circulation is maximum.

• The curl of a vector field B describes the rotational property.

• For a closed contour C, circulation=

• The curl is defined as

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Cd B l

0 max

1ˆcurl lim

Csd

s

B B n B l

The direction of the unit vector n is along the thumb when the other 4 fingers of the right hand follow dl

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Right-hand rule:

Curl In Cartesian Coordinate

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ˆ ˆ ˆ

ˆ ˆ ˆ

x y z

x y z

xB yB zB

x y z

x y z

B B B

B

B =

For Vector B,

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Curl is zero for uniform field

since there is no circulation.

Curl is non-zero for the

azimuthal field

Properties of the Curl

• x(A + B) =x A+x B ; A and B are vectors

• •( x A) = 0, the divergence of the curl of a vector field vanishes.

• x(V) = 0, the curl of the gradient of a scalar field vanishes.

• The curl of a vector field is another vector field;

• The curl of a scalar filed V, makes no sense.

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Stokes’s Theorem

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(s c

d d s B lx )B

If X B = 0, the field B is conservative, or

irrotational (as its circulation = 0)

Example 3-4: Verification of Stoke’s Theorem

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(s c

d d s B lx )B

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What is a Laplacian?

• Laplacian of a scalar function is defined as the divergence of the gradient of that function.

• Or we also can say: The divergence of the gradient of a scalar function is called the Laplacian.

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Laplacian Operator

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2 2 2

2 2 2

2 2 22

2 2 2

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

( )

( )

x y z

yx z

V V VV grad V x y z

x y z

A A A

x y z

AA AV

x y z

V V V

x y z

V V VV V

x y z

x y z A

x y z

A =

Recall the

gradient

And

Now take a

divergence

Laplacian is

defined as

Laplacian of a scalar function is defined as the

divergence of the gradient of that function.

For a vector field E

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2 2 22 2 2 2

2 2 2

2

ˆ ˆ ˆx y zE E E

x y z

E = E = x y z

E = ( E) - ( E)

With identity:

The definition of a scalar Laplacian can be used to

define a Laplacian of a vector filed

2 can also write as , call “del square”

or Laplacian.

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