RF/Microwave Engineering and Applications in Energy Systems. Abdullah Eroglu
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Название: RF/Microwave Engineering and Applications in Energy Systems

Автор: Abdullah Eroglu

Издательство: John Wiley & Sons Limited

Жанр: Техническая литература

Серия:

isbn: 9781119268819

isbn:

СКАЧАТЬ phi Baseline Over partial-differential phi EndFraction plus StartFraction partial-differential upper T Subscript z Baseline Over partial-differential z EndFraction equals StartFraction 1 Over r EndFraction StartFraction partial-differential Over partial-differential r EndFraction left-parenthesis r squared cosine phi right-parenthesis plus StartStartFraction partial-differential left-parenthesis left-parenthesis StartFraction z Over r EndFraction right-parenthesis sine phi right-parenthesis OverOver partial-differential z EndEndFraction 2nd Row equals 2 cosine phi plus StartFraction 1 Over r EndFraction sine phi EndLayout"/>

      1.3.4 Curl

      The curl of a vector is used to identify how much the vector v curls around a reference point. The curl of a vector is expressed as

      (1.62)nabla x upper T overbar equals Start 3 By 3 Determinant 1st Row 1st Column ModifyingAbove x With ampersand c period circ semicolon 2nd Column ModifyingAbove y With ampersand c period circ semicolon 3rd Column ModifyingAbove z With ampersand c period circ semicolon 2nd Row 1st Column StartFraction partial-differential Over partial-differential x EndFraction 2nd Column StartFraction partial-differential Over partial-differential y EndFraction 3rd Column StartFraction partial-differential Over partial-differential z EndFraction 3rd Row 1st Column upper T Subscript x Baseline 2nd Column upper T Subscript y Baseline 3rd Column upper T Subscript z Baseline EndDeterminant equals ModifyingAbove x With ampersand c period circ semicolon left-parenthesis StartFraction partial-differential upper T Subscript z Baseline Over partial-differential y EndFraction minus StartFraction partial-differential upper T Subscript y Baseline Over partial-differential z EndFraction right-parenthesis plus ModifyingAbove y With ampersand c period circ semicolon left-parenthesis StartFraction partial-differential upper T Subscript x Baseline Over partial-differential z EndFraction minus StartFraction partial-differential upper T Subscript z Baseline Over partial-differential x EndFraction right-parenthesis plus ModifyingAbove z With ampersand c period circ semicolon left-parenthesis StartFraction partial-differential upper T Subscript y Baseline Over partial-differential x EndFraction minus StartFraction partial-differential upper T Subscript x Baseline Over partial-differential y EndFraction right-parenthesis

      Curl can be given in a cylindrical or spherical coordinate system as

      (1.63)nabla x upper T overbar equals StartFraction 1 Over r EndFraction Start 3 By 3 Determinant 1st Row 1st Column ModifyingAbove r With ampersand c period circ semicolon 2nd Column r ModifyingAbove phi With ampersand c period circ semicolon 3rd Column ModifyingAbove z With ampersand c period circ semicolon 2nd Row 1st Column StartFraction partial-differential Over partial-differential r EndFraction 2nd Column StartFraction partial-differential Over partial-differential phi EndFraction 3rd Column StartFraction partial-differential Over partial-differential z EndFraction 3rd Row 1st Column upper T Subscript r Baseline 2nd Column italic r upper T Subscript phi Baseline 3rd Column upper T Subscript z Baseline EndDeterminant equals ModifyingAbove r With ampersand c period circ semicolon left-parenthesis StartFraction 1 Over r EndFraction StartFraction partial-differential upper T Subscript z Baseline Over partial-differential phi EndFraction minus StartFraction partial-differential upper T Subscript phi Baseline Over partial-differential z EndFraction right-parenthesis plus ModifyingAbove phi With ampersand c period circ semicolon left-parenthesis StartFraction partial-differential upper T Subscript r Baseline Over partial-differential z EndFraction minus StartFraction partial-differential upper T Subscript z Baseline Over partial-differential r EndFraction right-parenthesis plus ModifyingAbove z With ampersand c period circ semicolon StartFraction 1 Over r EndFraction left-parenthesis StartFraction partial-differential Over partial-differential r EndFraction left-parenthesis italic r upper T Subscript phi Baseline right-parenthesis minus StartFraction partial-differential upper T Subscript r Baseline Over partial-differential phi EndFraction right-parenthesis

      and

      (1.64)StartLayout 1st Row nabla x upper T overbar equals StartFraction 1 Over r squared sine theta EndFraction Start 3 By 3 Determinant 1st Row 1st Column ModifyingAbove upper R With ampersand c period circ semicolon 2nd Column upper R ModifyingAbove theta With ampersand c period circ semicolon 3rd Column upper R sine theta ModifyingAbove phi With ampersand c period circ semicolon 2nd Row 1st Column StartFraction partial-differential Over partial-differential upper R EndFraction 2nd Column StartFraction partial-differential Over partial-differential theta EndFraction 3rd Column StartFraction partial-differential Over partial-differential phi EndFraction 3rd Row 1st Column upper T Subscript r Baseline 2nd Column italic upper R upper T Subscript theta Baseline 3rd Column upper R sine theta upper T Subscript phi Baseline EndDeterminant equals ModifyingAbove upper R With ampersand c period circ semicolon StartFraction 1 Over upper R sine theta EndFraction left-parenthesis StartFraction partial-differential Over partial-differential theta EndFraction left-parenthesis upper T Subscript phi Baseline sine theta right-parenthesis minus StartFraction partial-differential upper T Subscript theta Baseline Over partial-differential phi EndFraction right-parenthesis plus ModifyingAbove theta With ampersand c period circ semicolon StartFraction 1 Over upper R EndFraction left-parenthesis StartFraction 1 Over sine theta EndFraction StartFraction partial-differential upper T Subscript upper R Baseline Over partial-differential phi EndFraction minus StartFraction partial-differential Over partial-differential upper R EndFraction left-parenthesis italic upper R upper T Subscript phi Baseline right-parenthesis right-parenthesis 2nd Row plus ModifyingAbove phi With ampersand c period circ semicolon StartFraction 1 Over upper R EndFraction left-parenthesis StartFraction partial-differential Over partial-differential upper R EndFraction left-parenthesis italic upper R upper T Subscript theta Baseline right-parenthesis minus StartFraction partial-differential upper T Subscript upper R Baseline Over partial-differential theta EndFraction right-parenthesis EndLayout

      It is important to note that curl operation on any vector results in another vector. Curl cannot operate on a scalar quantity. In addition, the following two properties follow for curl operation.

      (1.66)nabla x nabla upper T equals 0

      1.3.5 Divergence Theorem

      The divergence theorems states that the volume integral of the divergence of a vector is equal to the surface integral of the same vector enclosing that volume. It is mathematically given by