![]() ![]() And we've all seen water vapor.through a.water vapor in our bathroom when.you have a ray of sunlight, and you can see.how the particles.how the particles are traveling. especially when sunlight is shining through it. You know, you could imagine.something like water vapor in your bathroom. And this isn't like.some crazy, abstract thing. So this is.the flux through a 2D surface. This is essentially the flux through a two-dimensional surface. How much mass.is traveling across this surface.at any given moment in time?" And this is really the same idea we do with.the line integrals. we're essentially saying, "How much mass. and this is what essentially that surface integral is. this little.'infinitesimally' chunk of surface.in a given amount of time?" And then if we were to add up.all of the dS-es. this velocity, is going directly.out of this little dS. This is saying, "How much mass." "How much mass, given this mass density. Given how we've defined f.when we say what we say f represents. ![]() The units that we get for this.are kilogram per second. You have a meter and then.a meter squared in the numerator. And we're going to multiply that times meters squared. That's mass density.times the units of v.which is meters per second. So the units of f.are going to be the units of rho.which are going to be.kilogram per cubic meter. Let me write it in those colors.so we have.clear what's happening here. Let's say it could be.kilogram per meter-cubed. f's units are going to be.units of mass density, So it could be. n, right over here, just specifies.a direction. And this right over here.will essentially just give.the magnitude of that. ![]() "What is the magnitude of the.component of f that is.normal to the surface?" Or "How much of f is normal.to the surface?" So the component of f that is.normal to the surface.might look something like.might look something.like that. "What is." "What is the magnitude.of the component of f.that's going in the direction of n?" Or the component- Or. So what does all of this mean? Well when you take the dot product.of two vectors, this is essentially saying, "How much do they go together?" And since n is a unit vector. So right over here, f might look.f might look something like this. I'll get some f at any point.at any point in three-dimensional space. And f is defined throughout this three-dimensional space. The normal vector is going to.point right out of it. And I think when we do.particular units, it starts to make.a little bit more concrete sense. A dS is a little chunk of area.of that surface. It's completely analogous to.what we did in the two-dimensional case.with the line integrals. and hopefully that'll give us conceptual understanding.of what this thing right over here is measuring. And let's say my surface.I'll use that same color. And let's say that this right over here.is my y-axis. So let's think about what this is saying. where n is the unit normal vector at every. So we're going to evaluate over some surface. to evaluate the surface integral.over some surface. Now, what I want to do.is think about what it means.what it means.given this function, f. it'll make a little bit more conceptual sense. and we talk- think a little bit more.about them relative to a surface. Hopefully, as we use these two functions. you don't have to worry too much about it. One way to think about it is.kind of the momentum density. Obviously it maintains the direction of the velocity. And there's a couple of ways you could conceptualize this. Let me use the same color.that I used for v before. So for any point in (x,y,z).this will give us a vector, and then.we'll multiply it times this scalar right over here.for that same point in three dimensions. and it is equal to.the product.the product of rho and v. Now we're just extending it to three dimensions. We went through a very similar exercise.in two dimensions when we talked about line integrals. And this might all look a little bit familiar, because we did it. And this right over here tells us.the velocity of that same.the velocity of that same fluid or gas.or whatever we're talking about. It gives us a vector.for any point in three dimensions. And then, let's say we have another function, v. It just gives us a number.for any point in 3D. It gives us the mass density.at any point in three dimensions. Who knows what it is? Some type of substance. which is a function of (x,y,z).and it gives us the mass density.at any point in three dimensions. Please use this Sage link to check your work.Let's say we are operating.in three dimensions.
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