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Studying for a test? Prepare with these 13 lessons on Integrating multivariable functions.
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Let's say we are operating... ...in three dimensions. And I have a function, rho, ...which is a function of (x,y,z)... ...and it gives us the mass density... ...at any point in three dimensions, ...of some fluid. Some particular fluid. Maybe it's a gas, or a fluid, ...water. Who knows what it is? Some type of substance. It gives us the mass density... ...at any point in three dimensions. And let's say we have another... ...function. This is a scalar function. It just gives us a number... ...for any point in 3D. And then, let's say we have another function, v, ...which is a vector function. It gives us a vector... ...for any point in three dimensions. 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. Now let's imagine another function. And this might all look a little bit familiar, because we did it... We went through a very similar exercise... ...in two dimensions when we talked about line integrals. Now we're just extending it to three dimensions. Let's say we have a function, f. Let's say we have a function, f, ...and it is equal to... ...the product... ...the product of rho and v. 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. So it's equal to... ...rho times v. Let me use the same color... ...that I used for v before. And there's a couple of ways you could conceptualize this, ...so you could view this as... Obviously it maintains the direction of the velocity, ...but now its magnitude... One way to think about it is... ...kind of the momentum density. And if that doesn't make too much sense, ...you don't have to worry too much about it. Hopefully, as we use these two functions, ...and we talk-- think a little bit more... ...about them relative to a surface, ...it'll make a little bit more conceptual sense. Now, what I want to do... ...is think about what it means... ...what it means... ...given this function, f, ...to evaluate the surface integral... ...over some surface... So we're going to evaluate over some surface. We're going to evaluate f... We're going to evaluate f dot n, ...where n is the unit normal vector at every-- ...at each point on that surface, ...dS. d-surface. So let's think about what this is saying. So first, let me draw my axes. So I have my z-axis. z-axis... This could be my... Let's make that... Let's make that my x-axis. And let's say that this right over here... ...is my y-axis. And let's say my surface... ...I'll use that same color... My surface looks something like that. So that is my surface. That is the surface in question. That is S. Now let's think about the units, ...and hopefully that'll give us conceptual understanding... ...of what this thing right over here is measuring. It's completely analogous to... ...what we did in the two-dimensional case... ...with the line integrals. So we have a dS. A dS is a little chunk of area... ...of that surface. So that is dS. So this is going to be area. And if we want to break... This is... We want to pick particular units... This could be square... This could be square meters. And I think when we do... ...particular units, it starts to make... ...a little bit more concrete sense. Now, the normal vector... ...at that dS... The normal vector is going to... ...point right out of it. It's literally normal to that plane. It's literally normal to that plane. It has a magnitude 1. So that is our unit normal vector. And f is defined throughout this three-dimensional space. You give me any (x,y,z), ...I'll know its mass density, ...I'll know its velocity, ...and I'll get some f. I'll get some f at any point... ...at any point in three-dimensional space, ...including on the surface. Including right over here. So right over here, f might look... ...f might look something like this. So that is f right at that point. Right at that point. 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, ...since it has a magnitude 1, ...it's-- this is essentially saying.. "What is..." "What is the magnitude... ...of the component of f... ...that's going in the direction of n?" Or the component-- Or... "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. Might look something like that. And this right over here... ...will essentially just give... ...the magnitude of that. And it's just going to... It's just going to keep the units of f. n, right over here, just specifies... ...a direction. It has no units associated with it. It's dimensionless. f's units are going to be... ...units of mass density, So it could be... It's going to be... Let's say it could be... ...kilogram per meter-cubed. That's... Well that's actually just the rho part. So it's mass density times velocity. Times meters per second. Let me write it in those colors... ...so we have... ...clear what's happening here. So the units of f... ...are going to be the units of rho... ...which are going to be... ...kilogram per cubic meter... That's mass density. ...times the units of v... ...which is meters per second. Meters per second. And we're going to multiply that times meters squared. So what you have is... You have a meter and then... ...a meter squared in the numerator... That's meters cubed in the numerator. And meters cubed in the denominator. That, that, that cancels out. And so the units that we get for this... The units that we get for this... ...are kilogram per second. And so the way to conceptualize it... Given how we've defined f... ...when we say what we say f represents... The way to conceptualize this... This is saying, "How much mass..." "How much mass, given this mass density, ...this velocity, is going directly... ...out of this little dS, ...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, ...and this is what essentially that surface integral is, ...we're essentially saying, "How much mass, ...in kilograms per second, ...that's what we picked... 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. This is essentially the flux through a two-dimensional surface. So this is... ...the flux through a 2D surface. And this isn't like... ...some crazy, abstract thing. I mean, you could imagine... You know, you could imagine... ...something like water vapor in your bathroom. Water vapor in your bathroom. And I like to imagine that, ...because that's actually visible, ...especially when sunlight is shining through it. 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. And you see they have a certain density... ...at different points. And so you could imagine... You could imagine... You care about the surface... ...the surface of your... Maybe you have a window. Maybe you have a window in the bathroom. So you have a window. And so, if you were... If the surface was the window, ...and the window... And let's say the window's open, ...so it's kind of a... There's nothing physical there. It's just kind of a... ...a rectangular surface that... ...things can pass freely through, ...if-- and f was essentially... ...the mass density of the water vapor times... ...the velocity of the water vapor, ...then this thing right over here will essentially tell you... ...the mass of water vapor that is traveling... ...through that window at any given moment of time. Another way to think about it is... Imagine a... Imagine a river. And I'm going to conceptualize this river... ...as kind of a... ...just a section of the river. And I'm conceptualizing it... This is kind of... ...a river. Obviously this would be the surface... ...that we normally see. But obviously it has some depth. It's three-dimensional in nature. And so we would know the density. Maybe it's constant. You know the density, and you know the velocity... ...at any point. That's what f gives us. So that tells us... As we said, we could view that as the... ...momentum density at any given point in time. And maybe our surface is some type of a net. Our surface is some type of a net. And the net doesn't even have to be rectangular. It could be some weird-shaped net. But I'll do it in rectangular, just because it's... ...easier to draw. It's some type of net that in no way impedes the flow... ...of the fluid. Then once again, ...when you evaluate this integral, ...it would tell you the mass of fluid that is flowing... ...through that net... ...at any given moment of time. So hopefully this makes... ...a little bit of conceptual sense now. In the next few videos, ...we'll actually think about... ...how to... ...how to calculate this, ...and how we can actually represent it... ...in different ways.