The Del Operator & Field Operations
10/6/2020
Electromagnetics: Electromagnetic Field Theory The Del Operator & Field Operations
Outline
• Scalar vs. Vector Fields • The Del Operator ∇ • Gradient of a Scalar Field ∇𝑉 • Divergence of a Vector Field ∇ · 𝐴⃗ • Curl of a Vector Field ∇ 𝐴⃗ • Laplacian Operation ∇
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Scalar vs. Vector Fields
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Scalar Field Vs. Vector Field (2D)
Scalar Field 𝑓 𝑥, 𝑦, 𝑧 Vector Field 𝐹⃗ 𝑥, 𝑦, 𝑧
• Magnitude only • Magnitude • Direction Slide 4
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Scalar Field Vs. Vector Field (3D)
Scalar Field 𝑓 𝑥, 𝑦, 𝑧 Vector Field 𝐹⃗ 𝑥, 𝑦, 𝑧
• Magnitude only • Magnitude • Direction Slide 5
Isocontour Lines
Isocontour lines trace the paths of equal value. Closely space isocontours conveys that the function is varying rapidly.
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The Del Operator
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The Del Operator ∇
The del operator is the vector differential operator. It is sort of a 3D derivative. Even though is a Coordinate Del Operator vector, it is never System written as ∇. Cartesian aaaˆˆˆ x x yzyz 1 Cylindrical aaaˆˆˆ z z Derived from the Cartesian equation using coordinate 11 Spherical aaˆˆ a ˆtransformation. rrr rsin
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Summary of Vector Operations
Operation Input Output
Vector Addition & Subtraction Vectors Vector UV Dot ProductUV Vectors Scalar Cross ProductUV Vectors Vector Gradientf Scalar Function Vector Function DivergenceU Vector Function Scalar Function CurlU Vector Function Vector Function Scalar Laplacian2V Scalar Function Scalar Function Vector Laplacian2U Vector Function Vector Function
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Gradient of a Scalar Field
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Gradient of a Scalar Field (1 of 3)
Start with a scalar field 𝑓 𝑥, 𝑦 .
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Gradient of a Scalar Field (2 of 3) Plot the gradient ∇𝑓 𝑥, 𝑦, 𝑧 on top of 𝑓 𝑥, 𝑦, 𝑧 . The background color is the original scalar field 𝑓 𝑥, 𝑦, 𝑧 .
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Gradient of a Scalar Field (3 of 3)
The gradient will always be perpendicular to the isocontour lines.
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The Gradient ∇𝑉
The gradient calculates how rapidly, and in what direction, a scalar function is increasing.
Coordinate Gradient Operator System VVV Cartesian Vaˆˆˆ a a x x yzyz VVV1 Cylindrical Vaˆˆˆ a a z z VV11 V Spherical Vaˆˆ a a ˆ rrr rsin
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Algebra Rules for the Gradient
UV U V Sum/Difference Rule
UV U V V U Product Rule
UVUUV 2 Quotient Rule VV
VnVVnn1 Power Rule
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Properties of the Gradient 1. The gradient of a scalar function is a vector function. 2. The magnitude of V is the local maximum rate of change in V. 3. V points in the direction of maximum rate of change in V. 4. V at any point is perpendicular to the constant V surface that passes through that point. 5. V points toward increasing numbers in V.
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Divergence of a Vector Field
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Divergence of a Vector Field (1 of 2)
Start with the following vector field 𝐹⃗ 𝑥, 𝑦 .
Observe how the field seems to be converging to point in the upper left and diverging from a point in the low right.
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Divergence of a Vector Field (2 of 2)
The divergence is ∇ · 𝐹⃗ 𝑥, 𝑦 .
Divergence measures the tendency of a vector field to diverge from a point or converge to a point.
Divergence is a scalar quantity and is shown as the colored regions. Blue indicates negative divergence (convergence) and red indicates positive divergence.
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Divergence of a Vector Field (2D)
Vector Field 𝐴⃗ 𝑥, 𝑦 Divergence ∇ · 𝐴⃗ 𝑥, 𝑦
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Divergence ∇ · 𝐴⃗
The divergence of a vector field is a scalar field that measures the tendency of a vector field to diverge from a point or converge to a point.
Coordinate Gradient Operation System
A Ay A Cartesian A x z xyz
11 A A A Cylindrical A z z 2 11 rAr A sin 1A Spherical A rrr2 sin r sin
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Algebra Rules for Divergence
A BAB Distributive Rule
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Properties of Divergence ∇ · 𝐴⃗
1. The divergence of a vector function is a scalar function. 2. The divergence of a scalar field does not make sense. 3. The original vector field will point form larger to smaller numbers in the scalar field.
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Curl of a Vector Field
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Curl of a Vector Field (1 of 2)
Start with the following vector field 𝐹⃗ 𝑥, 𝑦 .
Observe how the field seems to be rotating around two different axes.
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Curl of a Vector Field (1 of 2)
The curl is ∇ 𝐹⃗ 𝑥, 𝑦 .
Curl measures the tendency of the vector field to circulate around an axis.
Curl is a vector quantity and is shown as the blue arrows. The magnitude of the curl conveys the strength of the circulation. The direction of the curl is the axis of the circulation.
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Curl of a Vector Field (2D)
Vector Field 𝐴⃗ 𝑥, 𝑦 Curl ∇ 𝐴⃗ 𝑥, 𝑦
y
x z
y
x z
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Curl
The curl of a vector function is a vector function that quantifies the tendency of the vector field to circulate around an axis. The magnitude of the curl conveys the strength of the circulation. The direction of the curl is the axis of the circulation.
Coordinate Curl Operation System
AA AAzzyyAAxx A aaaˆˆˆxyz Cartesian yz zx xy
11AAAA A A A zz aaˆˆ a ˆ Cylindrical z zz
1111ArAsin A AA rA A aaaˆˆˆ rr Spherical r r rrrrrsin sin
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Algebra Rules for Curl
A BAB Distributive Rule
A B A BB A B AA B
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Properties of Curl
1. The curl of a vector function is a vector function. 2. The curl of a scalar field does not make sense. 3. Curl follows the right‐hand rule. A 4. The divergence of the curl of a vector field is always zero. ∇ · ∇ 𝐴⃗ 0 5. The curl of the gradient of a scalar field is always zero. ∇ ∇𝑉 0 6. Note that ∇ · 𝐴⃗ 𝐴⃗ · ∇. ∇ · 𝐴⃗ calculates the ⃗ ⃗ derivative of 𝐴, whereas 𝐴 · ∇ sets up A a derivative operation that is scaled by 𝐴⃗.
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Laplacian Operation
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Scalar Laplacian ∇ 𝑉
The scalar Laplacian is defined as the divergence of the gradient. It is 2 sort of a measure of the tendency of the scalar function to form bowls. VV
Coordinate Scalar Laplacian Operation System 222VVV 2V Cartesian x222yz
22 2 11VVV Cylindrical V 22 2 z
2 2211VVV 1 Spherical Vr22 sin 222 rr r rsin r sin
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Visualization of Scalar Laplacian ∇ 𝑉
2 Vx, y Vx , y
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Vector Laplacian ∇ 𝐴⃗
The vector Laplacian is defined as the 2 AA A gradient of the divergence minus the curl.
Coordinate Scalar Laplacian Operation System 22 2 2 Cartesian A Aax ˆˆˆxyyzz Aa Aa
22 2 2 Cylindrical A Aaˆˆˆ Aa Aaz z
22 2 2 Spherical A Aarrˆˆˆ Aa Aa
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Visualization of Vector Laplacian ∇ 𝐴⃗
Axy, 2 Axy,
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Properties of the Laplacian
1. The Laplacian of a scalar function is a scalar function. 2. The scalar Laplacian arises when dealing with electric potential. 3. The Laplacian of a vector function is a vector function. 4. The vector Laplacian arises in the wave equation.
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Classification of Vector Fields
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Irrotational A vector field is irrotational if A 0
Examples of Irrotational Fields Gravity Electric potential
Irrotational Rotational
Consequences: 1. Irrotational fields must flow in essentially straight lines. 2. If A 0, then Ad 0 and the field is said to be conservative. L
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Solenoidal (Divergenceless) A vector field is solenoidal if A 0
Examples of Solenoidal Fields Fluid flows Magnetic fields
Consequences: 1. If the vector function does not diverge from any points, it must form loops. 2. If A 0, then Ads0 and the field will have zero net flux. S
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Laplacian Vector Fields
A vector field is Laplacian if it is both irrotational and solenoidal A 0 A 0
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