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ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. Gradient, Divergence and Curl: the Basics . We first consider the position vector, l : where x , y , and z are rectangular unit vectors. . Since the unit vectors for rectangular coordinates are constants, we have for d l :.

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Dr hugh blanton entc 3331 l.jpg

ENTC 3331

RF Fundamentals

Dr. Hugh Blanton

ENTC 3331



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  • The operator, del: constants, we have for Ñ is defined to be (in rectangular coordinates) as:

    • This operator operates as a vector.

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Gradient l.jpg
Gradient constants, we have for

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

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dT magnitude and direction of the maximum change of the function in space.

=

Ñ

×

ˆ

T

a

l

dl

Directional derivatives:

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Divergence are two ways for a vector to multiply another vector:

  • 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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Gauss' Theorem (Gau consider Gauss' theorem (sometimes called the divergence theorem). b’s Theorem

  • We start with:

Surface Areas

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  • The second part, gives the value of the change in vector A that cuts perpendicular through the surface.

    • 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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  • We thus can write: gives the value of the change in vector A that cuts perpendicular through the surface.

    • where the vector S is the surface area vector.

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  • So what? 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.

    • 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 any thoughts or obstacles:

 diverges from original direction

(degree of) divergence  0

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Now suppose they turn and speed up. any thoughts or obstacles:

 diverges from original direction

(degree of) divergence >> 0

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Current of water any thoughts or obstacles:

 No divergence from original direction

(degree of) divergence = 0

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Current of water any thoughts or obstacles:

 Divergence from original direction

(degree of) divergence ≠ 0

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+ any thoughts or obstacles:

E-field between two plates of a capacitor.

Divergenceless

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I any thoughts or obstacles:

b-field inside a solenoid is homogeneous and divergenceless.

divergenceless  solenoidal

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CURL any thoughts or obstacles:


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+ any thoughts or obstacles:

+

  • 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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  • Example any thoughts or obstacles:

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for any scalar function V.

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Stoke’s Theorem any thoughts or obstacles:

  • General mathematical theorem of Vector Analysis:

Closed boundary of that surface.

Any surface

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  • Given a vector field any thoughts or obstacles:

    • Verify Stoke’s theorem for a segment of a cylindrical surface defined by:

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z any thoughts or obstacles:

y

x

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The integral of over the specified surface any thoughts or obstacles: S is

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z any thoughts or obstacles:

c

d

b

y

x

a

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The surface any thoughts or obstacles: 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 any thoughts or obstacles:

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curl or rot any thoughts or obstacles:

  • place paddle wheel in a river

  • no rotation at the center

  • rotation at the edges

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  • the vector any thoughts or obstacles: 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 l.jpg
Electric Field Lines any thoughts or obstacles:

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

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