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Let's say we are operating...

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...in three dimensions.

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And I have a function, rho,

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...which is a function of (x,y,z)...

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...and it gives us the mass density...

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...at any point in three dimensions,

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...of some fluid.

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Some particular fluid.

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Maybe it's a gas, or a fluid,

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...water. Who knows what it is?

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Some type of substance.

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It gives us the mass density...

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...at any point in three dimensions.

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And let's say we have another...

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...function. This is a scalar function.

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It just gives us a number...

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...for any point in 3D.

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And then, let's say we have another function, v,

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...which is a vector function.

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It gives us a vector...

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...for any point in three dimensions.

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And this right over here tells us...

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...the velocity of that same...

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...the velocity of that same fluid or gas...

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...or whatever we're talking about.

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Now let's imagine another function.

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And this might all look a little bit familiar, because we did it...

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We went through a very similar exercise...

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...in two dimensions when we talked about line integrals.

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Now we're just extending it to three dimensions.

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Let's say we have a function, f.

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Let's say we have a function, f,

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...and it is equal to...

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...the product...

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...the product of rho and v.

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So for any point in (x,y,z)...

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...this will give us a vector, and then...

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...we'll multiply it times this scalar right over here...

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...for that same point in three dimensions.

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So it's equal to...

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...rho times v.

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Let me use the same color...

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...that I used for v before.

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And there's a couple of ways you could conceptualize this,

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...so you could view this as...

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Obviously it maintains the direction of the velocity,

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...but now its magnitude...

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One way to think about it is...

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...kind of the momentum density.

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And if that doesn't make too much sense,

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...you don't have to worry too much about it.

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Hopefully, as we use these two functions,

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...and we talk-- think a little bit more...

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...about them relative to a surface,

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...it'll make a little bit more conceptual sense.

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Now, what I want to do...

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...is think about what it means...

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...what it means...

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...given this function, f,

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...to evaluate the surface integral...

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...over some surface...

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So we're going to evaluate over some surface.

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We're going to evaluate f...

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We're going to evaluate f dot n,

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...where n is the unit normal vector at every--

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...at each point on that surface,

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...dS.

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d-surface.

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So let's think about what this is saying.

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So first, let me draw my axes.

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So I have my z-axis.

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z-axis...

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This could be my...

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Let's make that...

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Let's make that my x-axis.

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And let's say that this right over here...

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...is my y-axis.

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And let's say my surface...

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...I'll use that same color...

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My surface looks something like that.

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So that is my surface.

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That is the surface in question.

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That is S.

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Now let's think about the units,

85
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...and hopefully that'll give us conceptual understanding...

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...of what this thing right over here is measuring.

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It's completely analogous to...

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...what we did in the two-dimensional case...

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...with the line integrals.

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So we have a dS.

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A dS is a little chunk of area...

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...of that surface.

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So that is dS.

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So this is going to be area.

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And if we want to break...

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This is...

97
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We want to pick particular units...

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This could be square...

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This could be square meters.

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And I think when we do...

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...particular units, it starts to make...

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...a little bit more concrete sense.

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Now, the normal vector...

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...at that dS...

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The normal vector is going to...

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...point right out of it.

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It's literally normal to that plane.

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It's literally normal to that plane.

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It has a magnitude 1.

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So that is our unit normal vector.

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And f is defined throughout this three-dimensional space.

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You give me any (x,y,z),

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...I'll know its mass density,

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...I'll know its velocity,

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...and I'll get some f.

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I'll get some f at any point...

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...at any point in three-dimensional space,

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...including on the surface.

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Including right over here.

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So right over here, f might look...

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...f might look something like this.

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So that is f right at that point.

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Right at that point.

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So what does all of this mean?

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Well when you take the dot product...

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...of two vectors, this is essentially saying,

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"How much do they go together?"

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And since n is a unit vector,

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...since it has a magnitude 1,

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...it's-- this is essentially saying..

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"What is..."

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"What is the magnitude...

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...of the component of f...

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...that's going in the direction of n?"

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Or the component-- Or...

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"What is the magnitude of the...

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...component of f that is...

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...normal to the surface?"

139
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Or "How much of f is normal...

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...to the surface?"

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So the component of f that is...

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...normal to the surface...

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...might look something like...

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...might look something...

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...like that.

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Might look something like that.

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And this right over here...

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...will essentially just give...

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...the magnitude of that.

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And it's just going to...

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It's just going to keep the units of f.

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n, right over here, just specifies...

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...a direction. It has no units associated with it.

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It's dimensionless.

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f's units are going to be...

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...units of mass density,

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So it could be...

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It's going to be...

159
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Let's say it could be...

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...kilogram per meter-cubed.

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That's...

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Well that's actually just the rho part.

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So it's mass density times velocity.

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Times meters per second.

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Let me write it in those colors...

166
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...so we have...

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...clear what's happening here.

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So the units of f...

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...are going to be the units of rho...

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...which are going to be...

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...kilogram per cubic meter...

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That's mass density.

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...times the units of v...

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...which is meters per second.

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Meters per second.

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And we're going to multiply that times meters squared.

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So what you have is...

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You have a meter and then...

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...a meter squared in the numerator...

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That's meters cubed in the numerator.

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And meters cubed in the denominator.

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That, that, that cancels out.

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And so the units that we get for this...

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The units that we get for this...

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...are kilogram per second.

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And so the way to conceptualize it...

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Given how we've defined f...

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...when we say what we say f represents...

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The way to conceptualize this...

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This is saying, "How much mass..."

191
00:05:35,576 --> 00:05:37,399
"How much mass, given this mass density,

192
00:05:37,399 --> 00:05:39,704
...this velocity, is going directly...

193
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...out of this little dS,

194
00:05:41,822 --> 00:05:42,359
...this little...

195
00:05:42,359 --> 00:05:44,138
...'infinitesimally' chunk of surface...

196
00:05:44,138 --> 00:05:46,156
...in a given amount of time?"

197
00:05:46,156 --> 00:05:46,880
And then if we were to add up...

198
00:05:46,880 --> 00:05:48,130
...all of the dS-es,

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00:05:48,130 --> 00:05:50,055
...and this is what essentially that surface integral is,

200
00:05:50,055 --> 00:05:52,033
...we're essentially saying,

201
00:05:52,033 --> 00:05:53,270
"How much mass,

202
00:05:53,270 --> 00:05:54,606
...in kilograms per second,

203
00:05:54,606 --> 00:05:55,324
...that's what we picked...

204
00:05:55,324 --> 00:05:56,510
How much mass...

205
00:05:56,510 --> 00:05:59,091
...is traveling across this surface...

206
00:05:59,091 --> 00:06:01,721
...at any given moment in time?"

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00:06:01,721 --> 00:06:03,351
And this is really the same idea we do with...

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00:06:03,351 --> 00:06:04,287
...the line integrals.

209
00:06:04,287 --> 00:06:07,908
This is essentially the flux through a two-dimensional surface.

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00:06:07,908 --> 00:06:09,352
So this is...

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00:06:09,352 --> 00:06:12,605
...the flux through a 2D surface.

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00:06:18,559 --> 00:06:19,164
And this isn't like...

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00:06:19,164 --> 00:06:21,582
...some crazy, abstract thing.

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00:06:21,582 --> 00:06:22,762
I mean, you could imagine...

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00:06:22,762 --> 00:06:24,023
You know, you could imagine...

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00:06:24,023 --> 00:06:26,717
...something like water vapor in your bathroom.

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00:06:26,717 --> 00:06:27,876
Water vapor in your bathroom.

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00:06:27,876 --> 00:06:28,722
And I like to imagine that,

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...because that's actually visible,

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...especially when sunlight is shining through it.

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00:06:32,107 --> 00:06:33,908
And we've all seen water vapor...

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00:06:33,908 --> 00:06:34,579
...through a...

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00:06:34,579 --> 00:06:35,725
...water vapor in our bathroom when...

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00:06:35,725 --> 00:06:37,241
...you have a ray of sunlight, and you can see...

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...how the particles...

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...how the particles are traveling.

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And you see they have a certain density...

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...at different points.

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And so you could imagine...

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00:06:43,441 --> 00:06:44,526
You could imagine...

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00:06:44,526 --> 00:06:46,493
You care about the surface...

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00:06:46,493 --> 00:06:48,493
...the surface of your...

233
00:06:48,493 --> 00:06:50,394
Maybe you have a window.

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Maybe you have a window in the bathroom.

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So you have a window.

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00:06:53,885 --> 00:06:55,021
And so, if you were...

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If the surface was the window,

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00:06:57,058 --> 00:06:57,704
...and the window...

239
00:06:57,704 --> 00:06:58,777
And let's say the window's open,

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00:06:58,777 --> 00:06:59,639
...so it's kind of a...

241
00:06:59,639 --> 00:07:01,072
There's nothing physical there.

242
00:07:01,072 --> 00:07:02,031
It's just kind of a...

243
00:07:02,031 --> 00:07:03,899
...a rectangular surface that...

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00:07:03,899 --> 00:07:06,331
...things can pass freely through,

245
00:07:06,331 --> 00:07:08,014
...if-- and f was essentially...

246
00:07:08,014 --> 00:07:10,128
...the mass density of the water vapor times...

247
00:07:10,128 --> 00:07:11,670
...the velocity of the water vapor,

248
00:07:11,670 --> 00:07:13,677
...then this thing right over here will essentially tell you...

249
00:07:13,677 --> 00:07:15,687
...the mass of water vapor that is traveling...

250
00:07:15,687 --> 00:07:18,341
...through that window at any given moment of time.

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00:07:18,341 --> 00:07:19,774
Another way to think about it is...

252
00:07:19,774 --> 00:07:20,804
Imagine a...

253
00:07:20,804 --> 00:07:21,862
Imagine a river.

254
00:07:21,862 --> 00:07:23,695
And I'm going to conceptualize this river...

255
00:07:23,695 --> 00:07:24,606
...as kind of a...

256
00:07:24,606 --> 00:07:26,644
...just a section of the river.

257
00:07:26,644 --> 00:07:27,894
And I'm conceptualizing it...

258
00:07:27,894 --> 00:07:28,718
This is kind of...

259
00:07:28,718 --> 00:07:29,218
...a river.

260
00:07:29,218 --> 00:07:30,449
Obviously this would be the surface...

261
00:07:30,449 --> 00:07:31,573
...that we normally see.

262
00:07:31,573 --> 00:07:32,960
But obviously it has some depth.

263
00:07:32,960 --> 00:07:34,585
It's three-dimensional in nature.

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00:07:34,585 --> 00:07:36,838
And so we would know the density.

265
00:07:36,838 --> 00:07:37,686
Maybe it's constant.

266
00:07:37,686 --> 00:07:40,060
You know the density, and you know the velocity...

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00:07:40,060 --> 00:07:41,105
...at any point.

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00:07:41,105 --> 00:07:42,786
That's what f gives us.

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00:07:42,786 --> 00:07:44,558
So that tells us...

270
00:07:44,558 --> 00:07:46,789
As we said, we could view that as the...

271
00:07:46,789 --> 00:07:49,358
...momentum density at any given point in time.

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00:07:49,358 --> 00:07:51,939
And maybe our surface is some type of a net.

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00:07:51,939 --> 00:07:53,434
Our surface is some type of a net.

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And the net doesn't even have to be rectangular.

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It could be some weird-shaped net.

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But I'll do it in rectangular, just because it's...

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...easier to draw.

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It's some type of net that in no way impedes the flow...

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...of the fluid.

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Then once again,

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...when you evaluate this integral,

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...it would tell you the mass of fluid that is flowing...

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...through that net...

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...at any given moment of time.

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So hopefully this makes...

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...a little bit of conceptual sense now.

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In the next few videos,

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...we'll actually think about...

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...how to...

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...how to calculate this,

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...and how we can actually represent it...

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...in different ways.