RF/Microwave Engineering and Applications in Energy Systems. Abdullah Eroglu
Чтение книги онлайн.

Читать онлайн книгу RF/Microwave Engineering and Applications in Energy Systems - Abdullah Eroglu страница 24

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 target="_blank" rel="nofollow" href="#fb3_img_img_549360b5-53cb-52a4-8bea-819f5b24c543.png" alt="StartFraction italic d phi Over italic d s EndFraction equals ModifyingAbove nabla With bar phi dot StartFraction d r overbar Over italic d s EndFraction equals ModifyingAbove nabla With bar phi dot ModifyingAbove u With ampersand c period circ semicolon"/>

Schematic illustration of gradient.

      (1.56)StartFraction italic d phi Over italic d s EndFraction equals ModifyingAbove nabla With bar phi dot ModifyingAbove a With ampersand c period circ semicolon equals StartAbsoluteValue ModifyingAbove nabla With bar phi EndAbsoluteValue cosine theta

      in a cylindrical coordinate system as

      (1.57)nabla upper T equals ModifyingAbove r With ampersand c period circ semicolon StartFraction partial-differential upper T Over partial-differential r EndFraction plus ModifyingAbove phi With ampersand c period circ semicolon StartFraction 1 Over r EndFraction StartFraction partial-differential upper T Over partial-differential phi EndFraction plus ModifyingAbove z With ampersand c period circ semicolon StartFraction partial-differential upper T Over partial-differential z EndFraction

      and in a spherical coordinate system as

      (1.58)nabla upper T equals ModifyingAbove upper R With ampersand c period circ semicolon StartFraction partial-differential upper T Over partial-differential upper R EndFraction plus ModifyingAbove theta With ampersand c period circ semicolon StartFraction 1 Over upper R EndFraction StartFraction partial-differential upper T Over partial-differential theta EndFraction plus ModifyingAbove phi With ampersand c period circ semicolon StartFraction 1 Over upper R sine theta EndFraction StartFraction partial-differential upper T Over partial-differential phi EndFraction

      Example 1.2 Gradient

      If function ϕ = x2y + yz is given at point (1,2−1). (a) Find its rate of change for a distance in the direction r overbar equals ModifyingAbove x With ampersand c period circ semicolon plus 2 ModifyingAbove y With ampersand c period circ semicolon plus 3 ModifyingAbove z With ampersand c period circ semicolon period (b) What is the greatest possible rate of change with distance and in which direction does it occur at the same point?

      Solution

      1  Then, at point (1,2,−1)So, its rate of change is found fromwhere

      2 The greatest possible rate of change at (1,2,−1) is found fromwhere at (1,2,−1).

      1.3.3 Divergence

      The divergence of a vector field at a point is a measure of the net outward flux of the same vector per unit volume. The divergence of vector upper T overbar is defined as

      (1.59)StartLayout 1st Row nabla dot upper T overbar equals left-parenthesis ModifyingAbove x With ampersand c period circ semicolon StartFraction partial-differential Over partial-differential x EndFraction plus ModifyingAbove y With ampersand c period circ semicolon StartFraction partial-differential Over partial-differential y EndFraction plus ModifyingAbove z With ampersand c period circ semicolon StartFraction partial-differential Over partial-differential z EndFraction right-parenthesis dot left-parenthesis ModifyingAbove x With ampersand c period circ semicolon upper T Subscript x Baseline plus ModifyingAbove y With ampersand c period circ semicolon upper T Subscript y Baseline plus ModifyingAbove z With ampersand c period circ semicolon upper T Subscript z Baseline right-parenthesis 2nd Row equals StartFraction partial-differential upper T Subscript x Baseline Over partial-differential x EndFraction plus ModifyingAbove y With ampersand c period circ semicolon StartFraction partial-differential upper T Subscript y Baseline Over partial-differential y EndFraction plus ModifyingAbove z With ampersand c period circ semicolon StartFraction partial-differential upper T Subscript z Baseline Over partial-differential z EndFraction EndLayout

      The divergence can be represented in a cylindrical coordinate system as

      (1.60)nabla dot upper T overbar equals StartFraction 1 Over r EndFraction StartFraction partial-differential Over partial-differential r EndFraction left-parenthesis italic r upper T Subscript r Baseline right-parenthesis plus StartFraction 1 Over r EndFraction StartFraction partial-differential upper T Subscript phi Baseline Over partial-differential phi EndFraction plus StartFraction partial-differential upper T Subscript z Baseline Over partial-differential z EndFraction

      In a spherical coordinate system, the divergence is given as

      (1.61)nabla dot upper T overbar equals StartFraction 1 Over r squared EndFraction StartFraction partial-differential Over partial-differential r EndFraction left-parenthesis r squared upper T Subscript r Baseline right-parenthesis plus StartFraction 1 Over r sine theta EndFraction StartFraction partial-differential Over partial-differential theta EndFraction left-parenthesis upper T Subscript theta Baseline sine theta right-parenthesis plus StartFraction 1 Over r sine theta EndFraction StartFraction partial-differential upper T Subscript phi Baseline Over partial-differential phi EndFraction

      The divergence of a vector field gives a scalar result. In addition, the divergence of a scalar is not a valid operation.

      Example 1.3 Divergence

      In cylindrical coordinate system, it is given that

upper T overbar equals r cosine phi ModifyingAbove r With ampersand c period circ semicolon plus left-parenthesis StartFraction z Over r EndFraction right-parenthesis sine phi ModifyingAbove z With ampersand c period circ semicolon

      Calculate nabla dot upper T overbar.

      Solution

StartLayout 1st Row nabla dot upper T overbar equals StartFraction 1 Over r EndFraction StartFraction partial-differential Over partial-differential r EndFraction left-parenthesis italic r upper T Subscript r Baseline right-parenthesis plus StartFraction 1 Over r EndFraction StartFraction partial-differential upper T <a href=СКАЧАТЬ