Exterior Calculus: Theory and Cases

Vector Algebra on R2 and R3

Author(s): Carlos Polanco * .

Pp: 1-13 (13)

DOI: 10.2174/9789814998789121010007

* (Excluding Mailing and Handling)

Abstract

In this chapter, we introduce the main operators of Heaviside-Gibbs algebra: addition, subtraction, norm of vectors, as well as inner and cross product. From the point of view of Vector Calculus, we introduce the line and surface integrals, and the Green’s, Stokes’, and Gauss’ Theorems. The last section discusses the extension of this algebra in n-dimensional space. The examples are in plane and space.


Keywords: cross product: v×w, divergence of vector function, Gauss’ Theorem, Green’s Theorem, inner product: v ·w, limitations, line integral, norm: ||v||, normed vector space, rotational of vector function, scalar multiplication: av, Stokes’ Theorem, surface integral, vector addition: v + w, vector Subtraction: v−w

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      • 14. app/Models/UserAccess/UserAccess.php:784
      • 15. middleware::check_user_access:114
      • 16. vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:183
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Middleware/SubstituteBindings.php:50
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      • 16. app/Models/Subject.php:238
      • 17. app/Http/Controllers/ChapterController.php:1434
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      • 14. app/Models/Subject.php:238
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      • 16. app/Models/Subject.php:255
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      • 11. app/Models/ContentAccess.php:647
      • 12. app/Http/Controllers/ChapterController.php:1465
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      340μsalphaeurekaselec_live_10_06_2022ContentAccess.php#647
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      • 11. app/Models/ContentAccess.php:647
      • 12. app/Http/Controllers/ChapterController.php:1478
      • 13. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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      630μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1518
      • 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
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      630μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
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      • 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` = 3105 and `b`.`banner_from_date` <= 1739989059 and `b`.`banner_to_date` >= 1739989059 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.09msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 3105
      • 3: 1739989059
      • 4: 1739989059
      • 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='c23710ba6922bd4ee98a5d46e84b726d'
      250μsalphaeurekaselec_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\Keywords
    15Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
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        uc_cart_id
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        _flash
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        PHPDEBUGBAR_STACK_DATA
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        path_info
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        status_code
        200
        
        status_text
        OK
        format
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        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
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        request_cookies
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        response_headers
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        ClearShow all
        Date ↕MethodURLData
        #12025-02-19 18:17:39GET/chapter/153251121817