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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:
1
⋅ × = ⋅ × = ⋅ × (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
−
−= ()
−
15
= lim→0 1
2 exp
2