Unit 14: Flux in 3D

The Concept

Let us introduce the idea of flux with a typical application. We are given a vector field [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}  \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], and [latex]\vec{n}=\frac{∇g}{||∇g||}[/latex] is represents the unit normal vector and [latex]∇g=\langle \frac{∂g}{∂x},\frac{∂g}{∂y},\frac{∂g}{∂z} \rangle[/latex]. 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.

The Plot

Now, you should engage with the 3D plot below to understand flux[1]. Follow the steps below to apply changes to the plot and observe the effects:

  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).
  2. Input the surface function.
  3. The graph depicted shows the flux.

Self-Checking Questions

Check your understanding by solving the following question[2]:

  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].

Use the graph to find the answer to this question.

  1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.
  2. Gilbert Strang, Edwin “Jed” Herman,  OpenStax, Calculus Volume 3,  Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).


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3D Interactive Plots for Multivariate Calculus by Dr. Na Yu, Ryerson University is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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