{"id":295,"date":"2022-02-23T17:40:33","date_gmt":"2022-02-23T22:40:33","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=295"},"modified":"2024-01-30T23:39:45","modified_gmt":"2024-01-31T04:39:45","slug":"flux-in-3d","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/flux-in-3d\/","title":{"raw":"Unit 14: Flux in 3D","rendered":"Unit 14: Flux in 3D"},"content":{"raw":"

The Concept<\/h2>\r\nLet us introduce the idea of flux<\/strong>\u00a0with a typical application. We are given a vector field <\/strong>[latex]\\vec{F}= \\langle P(x, y,z) , Q(x, y,z) , R(x,y,z) \\rangle [\/latex] that represents the flow of a fluid, for example [latex]\\vec{F}[\/latex] represents the velocity of wind in 3D. The flux is the rate of the flow per unit time. The flux of [latex]\\vec{F}[\/latex] across surface [latex]S[\/latex] is the line integral denoted by [latex] \\int_{S}\u00a0 \\vec{F} \\cdot n(t)\\, ds [\/latex], where [latex]\\vec{F}[\/latex] is a vector field, surface [latex]S[\/latex] is defined by [latex]g(x,y,z) = 0[\/latex] with gradient vector [latex] \u2207g=\\langle \\frac{\u2202g}{\u2202x},\\frac{\u2202g}{\u2202y},\\frac{\u2202g}{\u2202z} \\rangle[\/latex], and\u00a0 [latex]\\vec{n}=\\frac{\u2207g}{||\u2207g||}[\/latex] represents the unit normal vector. Imagine surface [latex]S[\/latex] is a membrane across which fluid flows, but [latex]S[\/latex] does not impede the flow of the fluid. In other words, [latex]S[\/latex] is an idealized membrane invisible to the fluid. Suppose [latex]F[\/latex] represents the velocity field of the fluid.\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below to understand flux[footnote]Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.[\/footnote]. Follow the steps below to apply changes to the plot and observe the effects:\r\n
    \r\n \t
  1. Fill in [latex]P(x, y,z)[\/latex], [latex]Q(x, y,z)[\/latex] and [latex]R(x, y,z) [\/latex](i.e., three compartments of the vector field function).<\/li>\r\n \t
  2. Input the surface function.<\/li>\r\n \t
  3. The graph depicted shows the flux.<\/li>\r\n<\/ol>\r\n[h5p id=\"26\"]\r\n
    \r\n

    Self-Checking Questions<\/h2>\r\n<\/header>\r\n
    \r\n\r\nCheck your understanding by solving the following question[footnote]Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).[\/footnote]<\/span>:\r\n
      \r\n \t
    1. Consider the radial field [latex]\\vec{F}(x,y,z)= \\frac{\\langle x,y,z \\rangle}{(x^2+y^2+z^2)}[\/latex] and sphere [latex]S[\/latex] centred at the origin with radius 1. Find the total outward flux across [latex]S[\/latex].<\/li>\r\n<\/ol>\r\nUse the graph to find the answer to this question.\r\n\r\n<\/div>\r\n<\/div>","rendered":"

      The Concept<\/h2>\n

      Let us introduce the idea of flux<\/strong>\u00a0with a typical application. We are given a vector field <\/strong>[latex]\\vec{F}= \\langle P(x, y,z) , Q(x, y,z) , R(x,y,z) \\rangle[\/latex] that represents the flow of a fluid, for example [latex]\\vec{F}[\/latex] represents the velocity of wind in 3D. The flux is the rate of the flow per unit time. The flux of [latex]\\vec{F}[\/latex] across surface [latex]S[\/latex] is the line integral denoted by [latex]\\int_{S}\u00a0 \\vec{F} \\cdot n(t)\\, ds[\/latex], where [latex]\\vec{F}[\/latex] is a vector field, surface [latex]S[\/latex] is defined by [latex]g(x,y,z) = 0[\/latex] with gradient vector [latex]\u2207g=\\langle \\frac{\u2202g}{\u2202x},\\frac{\u2202g}{\u2202y},\\frac{\u2202g}{\u2202z} \\rangle[\/latex], and\u00a0 [latex]\\vec{n}=\\frac{\u2207g}{||\u2207g||}[\/latex] represents the unit normal vector. Imagine surface [latex]S[\/latex] is a membrane across which fluid flows, but [latex]S[\/latex] does not impede the flow of the fluid. In other words, [latex]S[\/latex] is an idealized membrane invisible to the fluid. Suppose [latex]F[\/latex] represents the velocity field of the fluid.<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the 3D plot below to understand flux[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Fill in [latex]P(x, y,z)[\/latex], [latex]Q(x, y,z)[\/latex] and [latex]R(x, y,z)[\/latex](i.e., three compartments of the vector field function).<\/li>\n
      2. Input the surface function.<\/li>\n
      3. The graph depicted shows the flux.<\/li>\n<\/ol>\n
        \n
        <\/div>\n<\/div>\n
        \n
        \n

        Self-Checking Questions<\/h2>\n<\/header>\n
        \n

        Check your understanding by solving the following question[2]<\/sup><\/a><\/span>:<\/p>\n

          \n
        1. Consider the radial field [latex]\\vec{F}(x,y,z)= \\frac{\\langle x,y,z \\rangle}{(x^2+y^2+z^2)}[\/latex] and sphere [latex]S[\/latex] centred at the origin with radius 1. Find the total outward flux across [latex]S[\/latex].<\/li>\n<\/ol>\n

          Use the graph to find the answer to this question.<\/p>\n<\/div>\n<\/div>\n

          1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0. ↵<\/a><\/li>
          2. Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA). ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":396,"menu_order":14,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-295","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/295","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/users\/396"}],"version-history":[{"count":24,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/295\/revisions"}],"predecessor-version":[{"id":646,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/295\/revisions\/646"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/295\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=295"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=295"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=295"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}