Let's now expand our knowledge of calculus
to the third dimension.
So first of all, just what does a function look like
in three dimensions?
And actually we'll go over the different types.
Because you can have a line in three dimensions, or kind of
a curve in three dimensions.
You can have a surface.
You could have a vector field.
There are different types of representations we'll see,
when we start working with three dimensions.
But I think the most intuitive-- and none of these
are directly intuitive-- I think you have to really be
able to visualize them.
But the most intuitive, to me at least, is a surface
in three dimensions.
And eventually, we can expand this into n dimensions.
But then it becomes very hard to visualize.
So we had our traditional x and y-axis before, but
now let's give another dimension of height.
Let's say that this is my x-axis-- and I'll draw
the positive quadrant.
That's my x-axis.
That's the y-axis, and that's the z-axis.
And the convention is to kind of follow the right hand rule,
where the x-axis-- taking the cross product of the x-axis
with the y-axis-- is equal to the z-axis.
What do I mean by that?
This is x-- those colors really don't go well together.
This is y.
This is z.
What do I mean by the cross product?
So if this is the unit vector in the x
direction-- so that's i.
Let's say this is a length of 1.
This is in the y-direction, so it's j.
Oops.
j, little cap.
And that cap just means it's a unit vector.
It's in y-direction, but it has a magnitude of 1.
And I'll use a different color for z.
z is up.
And the unit vector for there is k.
And this is just a convention, that i cross j is equal to k.
And that's just the convention, you know,
for drawing the x-axis.
Do we make increasing x pointing out this way?
Or do we make increasing x point inwards?
And this gives us the convention.
So i goes in this direction.
I'm trying to make sure I can do my hand properly.
So let me draw the cross product.
So if you take the first vector, put your index finger
in the direction of the first vector, middle finger in the
direction of the second vector, and your other fingers can
do what they need to do.
So this is going in the direction of i.
That is going in the direction of k.
Sorry, of j, right?
In the y-direction.
And then you have your palm of your thumb, and then your thumb
is going to open up in this direction.
Your thumb is going to point up, which is
the direction of k.
So that's just a-- good to know, where did that
convention come from?
This is kind of called a right-handed coordinate system.
But let's get to the meat and potatoes.
So how do we define a surface in three dimensions?
Well, we can define z as a function of x and y.
So let's do that.
And just the notation, z is equal to a function of x and y.
And all that means is that if I give you an x-value and a
y-value you get a z-value.
And so, I don't know, let's pick one.
z is equal to x plus y.
So this z is equal to, I don't know, x squared plus y.
So how do we plot the points on the surface?
And I'll show you, actually, computer generated surfaces
that are far more professional looking than anything I
could possibly try to draw.
So let's say, what's f of-- f of, I don't know, 2 comma 1.
Well, that would mean x is 2, so it's 2 squared plus 1.
Well, it equals 5.
And if we had to plot that point of the surface-- and
maybe I'll actually graph this one in a little bit-- we
go along the x-axis 2.
So 1, 2.
We go along the y-axis 1.
So if you take this point-- this is x is equal
to y is equal to 1.
And then we go and then we say, well, z is equal to 5.
So we can go up here, I don't know, we'd go up 5 units.
And we would plot that point.
And you would see, if you kept doing that, you
would plot a surface.
You'd plot a surface.
And let me clean this up a little bit.
So the natural question that you might want to ask--
and actually, let me show you a surface.
I'm afraid that when I manipulate this graph, it'll
slow down my computer and I'll start sounding
like I'm melting.
But I'll take that risk.
Just bear with me.
So here is a surface I use, using this Java applet grapher.
And it's actually free.
I'll give you the link for it.
But this surface right here, this is-- I'll actually show
you the graph of this.
It'll start taking the partial derivative.
Don't worry about this wall.
We'll get to this in a second.
But this is a function of x and y.
You can see this is the x-axis, the y-axis.
The height is the z-axis.
This'll probably really slow down my computer, but you
can actually rotate it.
Look at that.
I don't want to slow.
I don't want to slow things too down while I'm trying
to do my screen capture.
Anyway, I think you'd understand, where you pick an
x-point, you pick a y-point, and then z-- this surface right
here, without this line intersecting it-- this
surface right here, is a function of x and y.
So the question is, well, how do we apply
calculus to surfaces?
Because-- actually, let me bring that thing out again.
Because if you look at this surface, if you were to pick
any arbitrary point on this surface, and say, what is
the slope of that surface?
Well, it kind of has no meaning, because you have to
kind of pick a direction.
If you said, what is the slope of the tangent line?
Any point on this graph actually has an infinite
number of tangent lines.
I mean, think of it this way.
Take a bowl or something that maybe-- you know, like this.
And then take a, I don't know, a toothpick.
And make that toothpick tangent to the bowl, and you can see
that on any point on the bowl, you can just rotate
that toothpick around.
So you kind of have to pick the orientation of that toothpick.
So what we're going to learn is when you take a derivative in
three dimensions, you have to specify the direction that
you're taking the derivative in.
And this is why I actually drew this wall here.
This wall is the equation y is equal to 0.3.
So you can kind of view it.
Along this wall, y is a constant, right?
So if we assume that y is constant, then maybe we could
take just the derivative with respect to x.
So we would essentially take the slope of this
curve right here.
And let's figure out how to do it.
So first of all, what is the equation of this surface?
And I just picked one that they had on Wikipedia.
But the equation of that surface is-- and I'm going to
remove this now, so I don't sound like I'm melting.
The equation of that surface-- and let me just clear out
everything, just because we'll probably need the extra space.
Go back to the pen tool.
The equation is z is equal to x squared plus xy plus y squared.
So we said if we want to take a derivative, it's hard to-- you
know, you can't just say there is one derivative.
We have to pick a direction.
We have to hold everything else constant and take the
derivative with respect to just one variable.
And that is called the partial derivative.
I know it sounds fancy, but you'll see.
It's actually no harder than taking a regular derivative.
You just have to make sure you remember which variable is a
variable, and which one is a constant.
So let's say we wanted to hold y constant.
And we just say, for any constant y, how much does z
change with respect to x?
Then we take the partial derivative-- this
is the notation.
You can view it as a d with the top curled.
The partial derivative of z with respect to x.
It equals-- all we do is we take this expression-- we take
the derivative of x-- and we just assume that y
is some constant.
So what's the derivative of 2x with respect to x?
Well, it's just 2x.
What's the derivative of xy with respect to x?
Well, y is just a number.
It's just a constant.
Remember, we're taking an implicit derivative here.
y is just a constant.
So if you have some constant times x, the derivative of
that is just the constant.
Plus y.
And then what's the derivative of y squared with respect to x?
Well, we're assuming y squared is a constant.
It's just a number, right? y is just a number.
So the derivative of just the number with respect
to x is just 0.
So the derivative of that is 0.
So the partial derivative of z with respect to x is 2x plus y.
Now, what does that mean?
Well, that means if I were-- and actually, let me give you
a little notation, before I show you what that means.
Another way to write this exact same thing is if we wrote that
f of xy is equal to the same thing-- x squared plus xy plus
y squared-- the partial of f with respect to x we could
have written as this.
The partial derivative of f with respect to x-- and still
a function of x and y, right?
It still depends on what constant y you're using--
is equal to 2x plus y.
Anyway, I thought it's nice to see that notation.
Now, what does this mean?
Well, what is the slope of z with respect to x at, say,
when x is 1 and-- actually, let's pick smaller numbers.
When x is equal to, I don't know, when x is equal to 0.2
and y is equal to, I don't know, 0.3.
Well, we could use this.
The partial derivative of f, with respect to x, at the
point 0.2, 0.3 is equal to 2 times x-- that's 0.4--
plus y-- plus 0.3.
So the slope of this function with respect to x at the 0.2,
comma 0.3, is equal to 0.7.
Let's see if we can visualize that.
So that wall represents the line y is equal to 0.3.
And we want the slope at equal at-- x is equal to 0.2.
So this is x is 0.2, right here.
So the rate at which the height, or the rate at
which z is changing with respect to x, is 0.7.
So every time x increases 1, z will increase by 0.7.
So the slope is a little bit less than 1.
I think you see that, right?
The tangent right here is increasing with increasing
values of x, but a little bit less than 45 degrees.
Anyway, I'm all out of time.
See you in the next video.