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### Course: Multivariable calculus>Unit 4

Lesson 13: Flux in 3D

# Conceptual understanding of flux

Conceptual understanding of flux across a two-dimensional surface. Created by Sal Khan.

## Want to join the conversation?

• Why do we only care about the momentum normal to the surface? particles traveling at an angle are also passing through the surface?
• Certainly they are! But what we want is the quantity of particles (mass) escaping exactly away from a surface. And by this "exactly away" I mean particles traveling away perpendicularly (represented by the normal) to the plane of the surface. So, the only way to represent any arbitrary traveling direction is by projecting (dot product) that particular direction against to the normal of the surface. Just imagine you have a flat sheet of paper that represent a surface placed right in front of your eyes. Now imagine a particle crossing trough the sheet in the ascending direction, but with an angle with the plane, that is diagonally. The net velocity of that particle crossing away the sheet (to be consistent with the concept "escaping really away") has to be measured by taking the dot product with the normal. Hope this helps
• At the beginning. Am I wrong if I picture the function ρ(x,y,z) as a measure of temperature in a room instead? I find this a bit easier to imagine, don't know why.
• Absolutely not! Actually you see this in tasks a lot. For instant calculating the heat flux around an ice cube, where you would get a negative flux or around a fire where you would get a positive flux. ( Given that vector n points outwards of the area around the ice cube and the fire.)
• At , why is it flux through a 2d surface rather than flux through a 3d surface? Isn't S a three dimensional surface?
• S is a 3D surface. But I think Sal says it because he's talking about flux/particles that pass through a small area (dS). dS is an infinitesimal (flat) parallelogram area. A flat area only has 2 dimensions: height and width(or x and y). (with 3 dimensions it would be a volume)

But I get the feeling most people(including myself) would actually refer to this concept as "flux in 3D". Even the title says "flux in three dimensions".
• In why do you use two integral signs to integrate over the surface
• Because you need a double integral to integrate over the surface. A surface is 2 dimensional.
In some books the write just one integral sign, but they also think of it as a double integral. So it's just a shortcut.
• Is it possible to calculate Flux within a piece of music?
• I think yes, though you need to define the surface (and thus the area) over which you are calculating the flux. Different notes cause different air speeds and so you would have to calculate the speed at a particular point in the music.
• Why are we making up a weird F(momentum density) from v and p(mass density),and why are we multiplying by a normal vector?
• at , surface integral? Where does he introduce this? a link to the appropriate video would be helpful.
• how to draw tangent vector to a surface? is it possible?
(1 vote)
• If you calculate the tangent plane at any point on a surface, then any vector from that point on the surface to any other point in that tangent plane will be a vector tangent to the surface at that point.

Hope that helps.
(1 vote)
• I know what mass and density is but not mass density. What is that?
(1 vote)
• If you had an object like tube that has a mass, the density could vary across the object, and I believe that's all mass density refers to. So if you picture this ll\/ll as your tube, you can see the the top would have a lower mass density than at the bottom, because of the taper.
(1 vote)
• Starting at Sal says " this is FLUX through a 2D surface " and he repeats this again. WHY is this a 2D surface, when he has drawn a 3D surface, and the title of the video is "conceptual understanding of FLUX in three dimensions?
(1 vote)