Dr. Hugh Blanton

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ENTC 3331 RF Fundamentals

• Dr. Hugh Blanton
• ENTC 3331

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Gradient, Divergence and Curl the Basics
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• We first consider the position vector, l
• where x, y, and z are rectangular unit vectors.

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• Since the unit vectors for rectangular
  coordinates are constants, we have for dl

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• The operator, del Ñ is defined to be (in
  rectangular coordinates) as
• This operator operates as a vector.

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Gradient

• If the del operator, Ñ operates on a scalar
  function, f(x,y,z), we get the gradient 

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• We can interpret this gradient as a vector with
  the magnitude and direction of the maximum change
  of the function in space.
• We can relate the gradient to the differential
  change in the function 

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Directional derivatives
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• Since the del operator should be treated as a
  vector, there are two ways for a vector to
  multiply another vector
• dot product and
• cross product.

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Divergence

• We first consider the dot product
• The divergence of a vector is defined to be
• This will not necessarily be true for other unit
  vectors in other coordinate systems.

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• To get some idea of what the divergence of a
  vector is, we consider Gauss' theorem (sometimes
  called the divergence theorem).

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Gauss' Theorem (Gaubs Theorem

• We start with

Surface Areas
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• We can see that each term as written in the last
  expression gives the value of the change in
  vector A that cuts perpendicular through the
  surface.

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• For instance, consider the first term
• The first part
• gives the change in the x-component of A

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• The second part,
• gives the yz surface (or x component of the
  surface, Sx) where we define the direction of the
  surface vector as that direction that is
  perpendicular to its surface.

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• The other two terms give the change in the
  component of A that is perpendicular to the xz
  (Sy) and xy (Sz) surfaces.

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• We thus can write
• where the vector S is the surface area vector.

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• Thus we see that the volume integral of the
  divergence of vector A is equal to the net amount
  of A that cuts through (or diverges from) the
  closed surface that surrounds the volume over
  which the volume integral is taken.
• Hence the name divergence for

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• So what?
• Divergence literally means to get farther apart
  from a line of path, or
• To turn or branch away from.

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• Consider the velocity vector of a cyclist not
  diverted by any thoughts or obstacles

Goes straight ahead at constant velocity.
? (degree of) divergence ? 0
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Now suppose they turn with a constant velocity
? diverges from original direction (degree
of) divergence ? 0
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Now suppose they turn and speed up.
? diverges from original direction (degree
of) divergence gtgt 0
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Current of water
? No divergence from original direction
(degree of) divergence 0
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Current of water
? Divergence from original direction
(degree of) divergence ? 0
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?
E-field between two plates of a capacitor.
Divergenceless
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I
b-field inside a solenoid is homogeneous and
divergenceless.
divergenceless ? solenoidal
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CURL
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• Two types of vector fields exists

Electrostatic Field where the field lines are
open and there is circulation of the field flux.
Magnetic Field where the field lines are closed
and there is circulation of the field flux.
circulation (rotation) ? 0
circulation (rotation) 0
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• The mathematical concept of circulation involves
  the curl operator.
• The curl acts on a vector and generates a vector.

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• In Cartesian coordinate system

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• Example

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• Important identities

for any scalar function V.
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Stokes Theorem

• General mathematical theorem of Vector Analysis

Closed boundary of that surface.
Any surface
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• Given a vector field
• Verify Stokes theorem for a segment of a
  cylindrical surface defined by

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z
y
x
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• Note that has only one component

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The integral of over the specified
surface S is
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z
c
d
b
y
x
a
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The surface S is bounded by contour C abcd. The
direction of C is chosen so that it is compatible
with the surface normal by the right hand
rule.
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Curl
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curl or rot

• place paddle wheel in a river
• no rotation at the center
• rotation at the edges

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• the vector un is out of the screen
• right hand rule
• Ds is surface enclosed within loop
• closed line integral

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Electric Field Lines

• Rules for Field Lines
• Electric field lines point to negative charges
• Electric field lines extend away from positive
  charges
• Equipotential (same voltage) lines are
  perpendicular to a line tangent of the electric
  field lines