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 $\vec{F}= \langle P(x, y,z) , Q(x, y,z) , R(x,y,z) \rangle$ that represents the flow of a fluid, for example $\vec{F}$ represents the velocity of wind in 3D. The flux is the rate of the flow per unit time. The flux of $\vec{F}$ across surface $S$ is the line integral denoted by $\int_{S} \vec{F} \cdot n(t)\, ds$, where $\vec{F}$ is a vector field, surface $S$ is defined by $g(x,y,z) = 0$, and $\vec{n}=\frac{∇g}{||∇g||}$ is represents the unit normal vector and $∇g=\langle \frac{∂g}{∂x},\frac{∂g}{∂y},\frac{∂g}{∂z} \rangle$. Imagine surface $S$ is a membrane across which fluid flows, but $S$ does not impede the flow of the fluid. In other words, $S$ is an idealized membrane invisible to the fluid. Suppose $F$ 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 $P(x, y,z)$, $Q(x, y,z)$ and $R(x, y,z)$(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 $\vec{F}(x,y,z)= \frac{\langle x,y,z \rangle}{(x^2+y^2+z^2)}$ and sphere $S$ centred at the origin with radius 1. Find the total outward flux across $S$.

Use the graph to find the answer to this question.