Partial derivatives Video Lecture - Engineering Mathematics

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.