Units

# 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] with gradient vector [latex]∇g=\langle \frac{∂g}{∂x},\frac{∂g}{∂y},\frac{∂g}{∂z} \rangle[/latex], and [latex]\vec{n}=\frac{∇g}{||∇g||}[/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.

## 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:

- 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).
- Input the surface function.
- The graph depicted shows the flux.

## Self-Checking Questions

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

- 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.