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VECTOR FIELDS
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VECTOR FIELDS
A distribution of a vector
quantity in space
A vector field 𝐴⃗ is represented by
• Cartesian
𝐴⃗(𝑥, 𝑦, 𝑧)
• Cylindrical
𝐴⃗(𝑟, 𝜙, 𝑧)
• Cylindrical
𝐴⃗(𝑅, 𝜃, 𝜙)
𝐴⃗ 𝑥, 𝑦, 𝑧 = 𝐱 + 𝐲
𝐴⃗ 𝑥, 𝑦, 𝑧 = 𝑥𝐱 − 𝑦𝐲
𝐴⃗ 𝑟, 𝜙, 𝑧 = −𝐫
𝐴⃗ 𝑟, 𝜙, 𝑧 = 𝑟𝛟
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EXAMPLE 3.1 (SKETCHING A VECTOR FIELD)
Sketch the following vector field
𝐴⃗ 𝑥, 𝑦, 𝑧 = −𝐱𝑦 + 𝐲2𝑥
Evaluate the vector field at several points
(𝑥, 𝑦)
(1,1)
(3, −2)
𝐴⃗(𝑥, 𝑦)
𝐴⃗ 1, 1 = −𝐱 1 + 𝐲2 1 = −𝐱 + 𝐲2
𝐴⃗ 3, −2 = −𝐱 −2 + 𝐲2 3 = 𝐱2 + 𝐲6
(−1, −1)
𝐴⃗ −1, −1 = −𝐱 −1 + 𝐲2 −1 = 𝐱 − 𝐲2
(−2,1)
𝐴⃗ −2, 1 = −𝐱 1 + 𝐲2 −2 = −𝐱 − 𝐲4
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EXAMPLE 3.2 (SKETCHING A VECTOR FIELD)
Sketch the following vector field
𝐴⃗ 𝑟, 𝜙, 𝑧 = 𝐫𝑟
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SCALAR FIELDS
A distribution of a scalar quantity in space
A scalar field 𝑉 is represented by
• Cartesian
𝑉(𝑥, 𝑦, 𝑧)
• Cylindrical
𝑉(𝑟, 𝜙, 𝑧)
• Cylindrical
𝑉(𝑅, 𝜃, 𝜙)
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VECTOR CALCULUS
Differentiation of vector fields
Gradient
:
Divergence :
Curl
:
∇𝑉
∇ ⋅ 𝐴⃗
∇ × 𝐴⃗
(Scalar  Vector)
(Vector  Scalar)
(Vector  Vector)
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EXAMPLE 3.3
Gradient ∇𝑉
Gradient of a scalar field, ∇𝑉
Cartesian:
Calculate the gradient of the scalar field
𝑉(𝑥, 𝑦, 𝑧) = 𝑥 𝑦 + 𝑥𝑦𝑧
Cylindrical
Scalar
field
Spherical
∇𝑉 =
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑥+
𝑦+
𝑧̂
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
𝜕𝑉
𝑟̂ +
𝜙+
𝑧̂
𝜕𝑟
𝜌 𝜕𝜙
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
1 𝜕𝑉
∇𝑉 =
𝑅+
𝜃+
𝜙
𝜕𝑅
𝑅 𝜕𝜃
𝑅 sin 𝜃 𝜕𝜙
∇𝑉 =
𝑉
(scalar field)
∇𝑉 = 𝐱
=𝐱
+𝐲
(
+𝐳
)
(
+𝐲
)
+𝐳
(
)
= 𝐱 2𝑥𝑦 + 𝑦𝑧 + 𝐲 𝑥 + 𝑥𝑧 + 𝐳𝑥𝑦
Vector
field
∇𝑉
(vector field)
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EXAMPLE 3.4
Gradient ∇𝑉
Gradient of a scalar field, ∇𝑉
Cartesian:
Cylindrical
Calculate the gradient of the scalar field
𝑉 = 𝑟 𝑧 cos 2𝜙
Spherical
Scalar
field
∇𝑉 = 𝐫
=𝐫
+𝛟
(
∇𝑉 =
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑥+
𝑦+
𝑧̂
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
𝜕𝑉
𝑟̂ +
𝜙+
𝑧̂
𝜕𝑟
𝜌 𝜕𝜙
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
1 𝜕𝑉
∇𝑉 =
𝑅+
𝜃+
𝜙
𝜕𝑅
𝑅 𝜕𝜃
𝑅 sin 𝜃 𝜕𝜙
∇𝑉 =
+𝐳
)
+𝛟
(
)
+𝐳
(
)
= 𝐫2𝑟𝑧 cos 2𝜙 − 𝛟2𝑟𝑧 sin 2𝜙 + 𝐳𝑟 cos 2𝜙
Vector
field
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EXAMPLE 3.5
Gradient ∇𝑉
Gradient of a scalar field, ∇𝑉
Cartesian:
Calculate the gradient of the scalar field
𝑉 = 10𝑅 sin 𝜃 cos 𝜙
∇𝑉 =
Cylindrical
Scalar
field
𝜕𝑉
1 𝜕𝑉
𝜕𝑉
𝑟̂ +
𝜙+
𝑧̂
𝜕𝑟
𝜌 𝜕𝜙
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
1 𝜕𝑉
∇𝑉 =
𝑅+
𝜃+
𝜙
𝜕𝑅
𝑅 𝜕𝜃
𝑅 sin 𝜃 𝜕𝜙
∇𝑉 =
Spherical
∇𝑉 = 𝐑
=𝐑
+𝛉
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑥+
𝑦+
𝑧̂
𝜕𝑥
𝜕𝑦
𝜕𝑧
+𝛟
(
)
+𝛉
(
)
(
+𝛟
)
= 𝐑10 sin 𝜃 cos 𝜙 + 𝛉10 sin 2𝜃 cos 𝜙 − 𝛟10 sin 𝜃 sin 𝜙
Vector
field
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EXAMPLE 3.6
Divergence ∇ ⋅ 𝐴⃗
Divergence of a vector field, ∇ ⋅ 𝐴⃗
Cartesian:
Calculate the divergence of the vector field
𝐴⃗ = 𝐱𝑥 𝑦𝑧 + 𝐲𝑥𝑧
Cylindrical
Vector
field
𝐴 = 𝑥 𝑦𝑧
Spherical
𝜕𝐴
𝜕𝐴
𝜕𝐴
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
1 𝜕(𝑟𝐴 ) 1 𝜕𝐴
𝜕𝐴
∇ ⋅ 𝐴⃗ =
+
+
𝑟 𝜕𝑟
𝑟 𝜕𝜙
𝜕𝑧
∇ ⋅ 𝐴⃗ =
∇ ⋅ 𝐴⃗
1 𝜕(𝑅 𝐴 )
1 𝜕(𝐴 sin 𝜃)
1 𝜕𝐴
=
+
+
𝑅
𝜕𝑅
𝑅 sin 𝜃
𝜕𝜃
𝑅 sin 𝜃 𝜕𝜙
𝐴⃗
(vector field)
𝐴 = 𝑥𝑧
𝐴 =0
∇ ⋅ 𝐴⃗ =
=
(
= 2𝑥𝑦𝑧
+
)
+
(
Scalar
field
+
)
+
( )
∇ ⋅ 𝐴⃗
(scalar field)
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EXAMPLE 3.7
Divergence ∇ ⋅ 𝐴⃗
Cartesian:
Divergence of a vector field, ∇ ⋅ 𝐴⃗
𝜕𝐴
𝜕𝐴
𝜕𝐴
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
1 𝜕(𝜌𝐴 ) 1 𝜕𝐴
𝜕𝐴
∇ ⋅ 𝐴⃗ =
+
+
𝑟 𝜕𝑟
𝑟 𝜕𝜙
𝜕𝑧
∇ ⋅ 𝐴⃗ =
Cylindrical
Calculate the divergence of the vector field
𝐴⃗ = 𝐫𝑟 sin 𝜙 + 𝛟𝑟 𝑧 + 𝐳𝑧 cos 𝜙
𝐴 = 𝑟 sin 𝜙
Spherical
𝐴 = 𝑧 cos 𝜙
∇ ⋅ 𝐴⃗
1 𝜕(𝑅 𝐴 )
1 𝜕(𝐴 sin 𝜃)
1 𝜕𝐴
=
+
+
𝑅
𝜕𝑅
𝑅 sin 𝜃
𝜕𝜃
𝑅 sin 𝜃 𝜕𝜙
𝐴 =𝑟 𝑧
(
∇ ⋅ 𝐴⃗ =
=
)
( (
))
+
+
(
+
)
(
+
)
= 2𝑟 sin 𝜙 + 0 + cos 𝜙
= 2 sin 𝜙 + cos 𝜙
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EXAMPLE 3.8
Divergence∇ ⋅ 𝐴⃗
Divergence of a vector field, ∇ ⋅ 𝐴⃗
Cartesian:
Calculate the divergence of the vector field
1
𝐴⃗ = 𝐑 cos 𝜃 + 𝛉𝑅 sin 𝜃 cos 𝜙 + 𝛟 cos 𝜃
𝑅
Cylindrical
1
cos 𝜃
𝑅
𝐴 = 𝑅 sin 𝜃 cos 𝜙
𝐴 =
∇ ⋅ 𝐴⃗ =
(
)
=
=
𝐴 = cos 𝜃
(
+
)
∇ ⋅ 𝐴⃗
1 𝜕(𝑟 𝐴 )
1 𝜕(𝐴 sin 𝜃)
1 𝜕𝐴
=
+
+
𝑟
𝜕𝑟
𝑟 sin 𝜃
𝜕𝜃
𝑟 sin 𝜃 𝜕𝜙
+
((
+
+
Spherical
𝜕𝐴
𝜕𝐴
𝜕𝐴
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
1 𝜕(𝜌𝐴 ) 1 𝜕𝐴
𝜕𝐴
∇ ⋅ 𝐴⃗ =
+
+
𝜌 𝜕𝜌
𝜌 𝜕𝜙
𝜕𝑧
∇ ⋅ 𝐴⃗ =
((
)
))
+
)
(
+
(
)
)
= 0+
2𝑅 sin 𝜃 cos 𝜃 cos 𝜙 + 0
= 2 cos 𝜃 cos 𝜙
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EXAMPLE 3.9
Curl ∇ × 𝐴⃗
Curl of a vector field, ∇ × 𝐴⃗
Cartesian:
∇ × 𝐴⃗ =
𝜕𝐴
𝜕𝐴
−
𝑥
𝜕𝑦
𝜕𝑧
𝜕𝐴
𝜕𝐴
+
−
𝑦
𝜕𝑧
𝜕𝑥
𝜕𝐴
𝜕𝐴
+
−
𝑧̂
𝜕𝑥
𝜕𝑦
Cylindrical
∇ × 𝐴⃗ =
𝜕𝐴
1 𝜕𝐴
−
𝑟̂
𝑟 𝜕𝜙
𝜕𝑧
𝜕𝐴
𝜕𝐴
+
−
𝜙
𝜕𝑧
𝜕𝑟
1 𝜕(𝜌𝐴 ) 𝜕𝐴
+
−
𝑧̂
𝑟 𝜕𝑟
𝜕𝜙
Spherical
∇ × 𝐴⃗ =
𝜕(𝐴 sin 𝜃) 𝜕𝐴
1
−
𝑅
𝑅 sin 𝜃
𝜕𝜃
𝜕𝜙
𝜕(𝑅𝐴 )
1
1 𝜕𝐴
+
−
𝜃
𝑅 sin 𝜃 𝜕𝜙
𝜕𝑅
1 𝜕(𝑅𝐴 ) 𝜕𝐴
+
−
𝜙
𝑅
𝜕𝑅
𝜕𝜃
𝐴 = 𝑥𝑦
𝐴 =𝑦
Calculate the curl of the vector field
𝐴⃗ = 𝐱𝑥𝑦 + 𝐲𝑦 + 𝐳𝑦𝑧
𝐴 = 𝑦𝑧
Vector
field
𝐴⃗
(vector field)
∇ × 𝐴⃗ = 𝐱
=𝐱
(
)
−
( )
−
+𝐲
+𝐲
(
−
)
−
+𝐳
(
)
+𝐳
−
( )
(
−
)
=𝐱 𝑧−0 +𝐲 0−0 +𝐳 0−𝑥
= 𝐱𝑧 − 𝐳𝑥
Vector
field
∇ × 𝐴⃗
(vector field)
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EXAMPLE 3.10
Curl ∇ × 𝐴⃗
Curl of a vector field, ∇ × 𝐴⃗
Calculate the curl of the vector field
𝑄 = 𝐫𝑟 sin 𝜙 + 𝛟𝑟 𝑧 + 𝐳𝑧 cos 𝜙
Cartesian:
∇ × 𝐴⃗ =
𝜕𝐴
𝜕𝐴
−
𝑎
𝜕𝑦
𝜕𝑧
𝜕𝐴
𝜕𝐴
+
−
𝑎
𝜕𝑧
𝜕𝑥
𝜕𝐴
𝜕𝐴
+
−
𝑎
𝜕𝑥
𝜕𝑦
Cylindrical
∇ × 𝐴⃗ =
𝜕𝐴
1 𝜕𝐴
−
𝑎
𝜌 𝜕𝜙
𝜕𝑧
𝜕𝐴
𝜕𝐴
+
−
𝑎
𝜕𝑧
𝜕𝜌
1 𝜕(𝜌𝐴 ) 𝜕𝐴
+
−
𝑎
𝜌
𝜕𝜌
𝜕𝜙
Spherical
∇ × 𝐴⃗ =
𝜕(𝐴 sin 𝜃) 𝜕𝐴
1
−
𝑎
𝑟 sin 𝜃
𝜕𝜃
𝜕𝜙
𝜕(𝑟𝐴 )
1 1 𝜕𝐴
+
−
𝑎
𝑟 sin 𝜃 𝜕𝜙
𝜕𝑟
1 𝜕(𝑟𝐴 ) 𝜕𝐴
+
−
𝑎
𝑟
𝜕𝑟
𝜕𝜃
𝑄 = 𝑟 sin 𝜙
𝑄 =𝑟 𝑧
𝑄 = 𝑧 cos 𝜙
∇ × 𝐴⃗ = 𝐫
=𝐫
(
−
)
−
+𝛟
(
)
+𝛟
−
(
+𝐳
)
= 𝐫 − 𝑧 sin 𝜙 − 𝑟
+𝛟 0−0 +𝐳
= −𝐫
+ 𝐳 3𝑟𝑧 − cos 𝜙
𝑧 sin 𝜙 + 𝑟
−
(
(
)
)
−
+𝐳
( (
))
−
3𝑟 𝑧 − 𝑟 cos 𝜙
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IMPORTANT PROPERTIES OF VECTOR FIELDS
Conservative
∇ × 𝐴⃗ = 0
All electrostatic fields
are conservative
Solenoidal
All magnetic fields are
solenoidal
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PHYSICAL MEANING OF DIVERGENCE
The divergence of a vector field is a measure of the "outwardness" or "inwardness" of the field at a
given point. More specifically, it measures how much the field is flowing out of or into a small region
around that point.
𝑃
𝑃
𝑃
There is a net outward flow of the
field at point P
∇ ⋅ 𝐴⃗ > 0
There is a net inward flow of the
field at point P
∇ ⋅ 𝐴⃗ < 0
There is a net flow of the field at
point P
∇ ⋅ 𝐴⃗ = 0
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PHYSICAL MEANING OF CURL
The curl of a vector field is a measure of the "circulation" or "rotation" of the field at a given point.
More specifically, it measures the tendency of the field to "curl around" a small region around that
point.
𝑃
𝑃
The field tends to rotate the paddle
counter-clockwise
∇ × 𝐴⃗ > 0
𝑃
The field does not tend to rotate the
paddle
∇ × 𝐴⃗ = 0
The field tends to rotate the paddle
clockwise
∇ × 𝐴⃗ < 0
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