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Semester II, 2015-16

Department of Physics, IIT Kanpur

PHY103A: Lecture # 3(Text Book: Intro to Electrodynamics by Griffiths, 3rd  Ed.) 

Anand Kumar Jha

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Triple Products:

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⋅ × =  ⋅ × = ⋅ ×  (2) ×  × =  ⋅ ( ⋅ ) 

Products Rules:

(3)  fg = f    + g   (4)  ⋅ =  ×  × + ×  × +  ⋅ +  ⋅  

(5)  ⋅ f  = f   ⋅ + ⋅ ( ) (6)  ⋅ × =  ⋅ × ⋅ ( × ) (7)  × f  = f   × × ( ) (8)

 ×

 ×

=

  ⋅ ⋅ +

⋅ (

⋅ ) 

Second Derivatives:(1)  ⋅ × = 0 (2)  ×   = 0 

(3)  ×  × =  ⋅ 2  2

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Integral Calculus:

Line Integral:

The ordinary integral that we know of is of the form: ∫   In vector calculus we encounter many other types of integrals.

� ⋅  

Vector field Infinitesimal

Displacement vector

Or

Line element

⋅  Closed path

=    + +  

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Example (G: Ex. 1.6)

Q: =    + 2( + 1) ? What is the line integralfrom A to B along path (1) and (2)?

∮ ⋅ =11-10=1

 Along path (1) We have =    + .(i) =  ; =1; ∫ ⋅ =∫ = 1 (ii) =  ; =2; ∫ ⋅ =∫ 4( + 1)1 =10

 Along path (2): =     + ; = ; =  ∫ ⋅ =∫ ( + 2 + 2)1 =10

Line Integral:

� ⋅

  ⋅  Closed path

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Surface Integral:

� ⋅ 

Vector field Infinitesimal

area vector

⋅ Closed path

• For a closed surface, the area vector points outwards.

• For open surfaces, the direction of area vector is decided based on a given problem.

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Surface Integral:

� ⋅  ⋅ Closed path

Example (Griffiths: Ex. 1.7)

Q: = 2    +  + 2  +   3 ? Calculate theSurface integral. 

(i) x=2, =  ; V ⋅ = 2 = 4 ∫ ⋅ = ∫ ∫ 400 =16

(ii) ∫ ⋅=0(iii) ∫ ⋅=12(iv)

∫ ⋅=-12

(v) ∫ ⋅=4(vi) ∫ ⋅=-12

⋅ = 1 6 + 0 + 1 2 1 2 + 4 1 2 = 8 6

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Volume Integral:

�(, , ) • In Cartesian coordinate system the volume

element is given by = . • One can have the volume integral of a vector

function which V as

∫= 

∫ 

+

∫+

∫ 

 

Example (Griffiths: Ex. 1.8)

Q: Calculate the volume integral of the =  over the volume of the prism

Q: We find that  integral runs from 0 to 3. The  integralruns from 0 to 1, but the  integral runs from 0 to 1  only.Therefore, the volume integral is given by

� � � 3

0

1

0=

3

8 

1−

0 

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Example

�  = � 

2

    =

2

 

  =

 

2  

•

In vector calculus, we have three different types of derivative (gradient, divergence, and curl) andcorrespondingly three different types of regions and end points.

The fundamental Theorem of Calculus:

� = � =  ()

 

The integral of a derivative

over a region is given by the

value of the function at the

boundaries

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The fundamental Theorem of Calculus:

� = � =  ()

 

The integral of a derivative

over a region is given by the

value of the function at the

boundaries

The fundamental Theorem for Gradient:

� ⋅ =  ()

     

The integral of a derivative

(gradient) over a  region (path) is

given by the value of the function at

the boundaries (end-points)

The fundamental Theorem for Divergence

(Gauss’s theorem or Divergence theorem):

The integral of a derivative

(divergence) over a region (volume)

is given by the value of the function at

the boundaries (bounding surface)

� ⋅ =  ⋅

 

 

� × ⋅ =  ⋅

 

 The integral of a derivative

(curl) over a region (surface) is

given by the value of the function

at the boundaries (closed-path)

The fundamental Theorem for Curl (Stokes’ theorem):

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Spherical Polar Coordinates:

=  sin cos , = sin sin, =  cos 

= A + A + A  A vector in the spherical polar coordinate is given by

= sin cos  + sin sin + cos  = cos cos  + cos sin   sin  = sin  + cos 

d l=  + +  The infinitesimal displacement vector in the spherical polar coordinate is given by

Griffiths: Prob 1.37

(In Cartesian system we have =    + +  )

0 ≤ ≤ ∞;0 ≤ ≤ ;0 ≤ ≤ 2 

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Spherical Polar Coordinates:

=  sin cos , = sin sin, =  cos 

= A + A + A  A vector in the spherical polar coordinate is given by

= sin cos  + sin sin + cos  = cos cos  + cos sin   sin  = sin  + cos 

d l=  + +  The infinitesimal displacement vector in the spherical polar coordinate is given by

Griffiths: Prob 1.37

(In Cartesian system we have =    + +  )d l=  + + sin  

The infinitesimal volume element: =  = sin 

The infinitesimal area element (it depends): =  = sin (over the surface of a sphere)

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Cylindrical Coordinates:

= s cos, =  sin, =  

= A + A + A A vector in the cylindrical coordinates is given by

= cos  + sin  

=   = sin  + cos 

d l=  + +  The infinitesimal displacement vector in the cylindrical coordinates is given by

Griffiths: Prob 1.41

(In Cartesian system we have =    + +  )

0 ≤ ≤ ∞;0 ≤ ≤ 2; ∞ ≤ ≤ ∞ 

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Cylindrical Coordinates:

= s cos, =  sin, =  

= A + A + A A vector in the cylindrical coordinates is given by

= cos  + sin  

=   = sin  + cos 

d l=  + +  The infinitesimal displacement vector in the cylindrical coordinates is given by

Griffiths: Prob 1.41

(In Cartesian system we have =    + +  )d l=  + +  

The infinitesimal volume element:

The infinitesimal area element (it depends):

=  =   (over the surface of a cylinder)

=  =   

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Dirac Delta function:

• Dirac delta function is a special function, which is defined as:

= 0,   ≠ 0= ∞,   = 0 

• Dirac delta function centered at

=

  is defined as follows

= 0,   ≠ 0= ∞,   =  

• If () is a continuous function of  

� = 1−  

� = 1−

 

� (

)

=

 � (

)

=

 (

)

−

− 

Also,  = 1 (),

• 3D Dirac delta function is defined as: 

3  =    � � � 3 

− 

−= ()

− 

 

 

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= lim→0  1

2 exp    

2