Field, Force, Energy and Momentum in Classical Electrodynamics (Revised Edition)

Mathematical Preliminaries

Author(s): Masud Mansuripur

Pp: 75-97 (23)

DOI: 10.2174/9781681085562117010006

* (Excluding Mailing and Handling)

Abstract

In preparation for a Fourier analysis of Maxwell’s equations in the following chapter, we describe here the mathematics of Fourier transformation, exploring certain properties of the forward and reverse Fourier operators. Several special functions are also discussed—notable among them, Dirac’s delta-function and various Bessel functions—which appear frequently in Fourier analysis and elsewhere. Simple charge- and current-density distributions serve as exemplary electromagnetic systems that can be readily transformed into the Fourier domain.

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      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Trial Access" as access_base_text, CASE j.type WHEN 111 THEN true ELSE false END as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_trail` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 873362154 and j.to_ip >=873362154 )) and ((j.from_date <= 1740457227 and j.to_date >= 1740457227 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 10896) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Subscribed" as access_base_text, false as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_corporate` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 873362154 and j.to_ip >=873362154 )) and ((j.from_date <= 1740457227 and j.to_date >= 1740457227 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 10896) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Token Based Access" as access_base_text, false as terms_popup, "R" as content_download_type, j.uc_id as restricted_user_access_key, "v" as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_token` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 873362154 and j.to_ip >=873362154 )) and ((j.from_date <= 1740457227 and j.to_date >= 1740457227 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 10896)
      690μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
      Backtrace
      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 2258 and `b`.`banner_from_date` <= 1740457227 and `b`.`banner_to_date` >= 1740457227 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.15msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 2258
      • 3: 1740457227
      • 4: 1740457227
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='68a129d05ab5c572381752761157df96'
      230μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Keywords
    1Keywords.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        ztyHdKHvTJAtUi1qBs1yb2NeiyFfrzbo9zgtv6xG
        uc_cart_id
        68a129d05ab5c572381752761157df96
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/10896" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/10896
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
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        request_cookies
        0 of 0
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        response_headers
        0 of 0
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        session_attributes
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        array:5 [ "_token" => "ztyHdKHvTJAtUi1qBs1yb2NeiyFfrzbo9zgtv6xG" "uc_cart_id" => "68a129d05ab5c572381752761157df96" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/10896" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 04:20:27GET/chapter/10896112193