Partial derivatives
Partial Derivatives Introduction to partial derivatives.
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- 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 not 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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