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

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in a spherical coordinate system.

      (1.30)ModifyingAbove upper A With right-arrow dot ModifyingAbove upper B With right-arrow equals upper A Subscript upper R Baseline upper B Subscript upper R Baseline plus upper A Subscript theta Baseline upper B Subscript theta Baseline plus upper A Subscript phi Baseline upper B Subscript phi

      (1.31)ModifyingAbove upper A With right-arrow times ModifyingAbove upper B With right-arrow equals Start 3 By 3 Determinant 1st Row 1st Column ModifyingAbove upper R With ampersand c period circ semicolon 2nd Column ModifyingAbove theta With ampersand c period circ semicolon 3rd Column ModifyingAbove phi With ampersand c period circ semicolon 2nd Row 1st Column upper A Subscript upper R Baseline 2nd Column upper A Subscript theta Baseline 3rd Column upper A Subscript phi Baseline 3rd Row 1st Column upper B Subscript upper R Baseline 2nd Column upper B Subscript theta Baseline 3rd Column upper B Subscript phi EndDeterminant

      (1.32)StartLayout 1st Row ModifyingAbove upper R With ampersand c period circ semicolon times ModifyingAbove theta With ampersand c period circ semicolon equals ModifyingAbove phi With ampersand c period circ semicolon 2nd Row ModifyingAbove theta With ampersand c period circ semicolon times ModifyingAbove phi With ampersand c period circ semicolon equals ModifyingAbove upper R With ampersand c period circ semicolon 3rd Row ModifyingAbove phi With ampersand c period circ semicolon times ModifyingAbove upper R With ampersand c period circ semicolon equals ModifyingAbove theta With ampersand c period circ semicolon EndLayout

      1.2.3 Differential Length (dl ), Differential Area (ds), and Differential Volume (dv )

      In vector analysis, line, surface, and volume integrals are expressed using differential lengths, areas, and volumes.

      1.2.3.1 dl, ds, and dv in a Cartesian Coordinate System

      Differential length represents infinitesimal change in any direction of the axis in the coordinate system and is represented by d ModifyingAbove l With right-arrow. In a Cartesian coordinate system, the differential length is given by

      (1.33)d ModifyingAbove l With right-arrow equals italic d x dot ModifyingAbove a With ampersand c period circ semicolon x plus italic d y dot ModifyingAbove a With ampersand c period circ semicolon y plus italic d z dot ModifyingAbove a With ampersand c period circ semicolon z

      Its magnitude is defined by

      (1.34)italic d l equals StartRoot italic d x squared plus italic d y squared plus italic d z squared EndRoot

      (1.35)StartLayout 1st Row d ModifyingAbove s With right-arrow Subscript x Baseline equals ModifyingAbove x With ampersand c period circ semicolon italic dydz 2nd Row d ModifyingAbove s With right-arrow Subscript y Baseline equals ModifyingAbove y With ampersand c period circ semicolon italic dxdz 3rd Row d ModifyingAbove s With right-arrow Subscript z Baseline equals ModifyingAbove z With ampersand c period circ semicolon italic dxdy EndLayout

      The infinitesimal change for the volume is defined by

      (1.36)italic d v equals italic dxdydz

      The illustration of the changes in length, area, and volume for a Cartesian coordinate system are given in Figure 1.9.

      1.2.3.2 dl, ds, and dv in a Cylindrical Coordinate System

      In a cylindrical coordinate system, the differential length is given by

      (1.37)d ModifyingAbove l With right-arrow equals italic d r dot ModifyingAbove a With ampersand c period circ semicolon r plus r dot italic d phi dot ModifyingAbove a With ampersand c period circ semicolon Subscript phi Baseline plus italic d z dot ModifyingAbove a With ampersand c period circ semicolon z

      Its magnitude is defined by

      (1.38)italic d l equals StartRoot italic d r squared plus r squared d phi squared plus italic d z squared EndRoot

Schematic illustration of differential length, area, and volume in a Cartesian coordinate system.

      (1.39)StartLayout 1st Row d ModifyingAbove s With right-arrow Subscript r Baseline equals ModifyingAbove r With ampersand c period circ semicolon italic rd phi dz 2nd Row d ModifyingAbove s With right-arrow Subscript phi Baseline equals ModifyingAbove phi With ampersand c period circ semicolon italic drdz 3rd Row d ModifyingAbove s With right-arrow Subscript z Baseline equals ModifyingAbove z With ampersand c period circ semicolon italic rdrd phi EndLayout

      The infinitesimal change for the volume is defined by

      (1.40)italic d v equals italic rdrd phi dz

      The illustration of the changes in length, area, and volume for a cylindrical coordinate system is given in Figure 1.10.

      1.2.3.3 dl, ds, and dv in a Spherical

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