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Let's learn a little bit about the gradient, and we'll use the

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same function we've been using since we're pretty familiar

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with it's graph and it's partial derivatives.

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So let's say that f of x y is equal to x squared plus x

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times y plus y squared.

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Now we're going to take the gradient of this, and then I'll

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give you a little bit of intuition on what

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a gradient is.

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We're going to take the gradient in two-dimensional

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space, and you'll see what that means in a second.

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You can extend this to any dimension.

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So the gradient that we use this vector differential

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operator-- this upside down triangle-- the gradient of f is

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equal to the partial of f in the x direction times the unit

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vector i-- which is the unit vector in the x direction so it

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gives the magnitude of the slope and the x direction, and

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then it points the vector in that direction-- plus the

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partial with respect to y in the j direction.

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And obviously if there's a function of more variables you

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could keep going and you'd multiply the magnitude of the

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derivative in each of those dimension's directions times

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the unit vector in that dimension.

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Well what does this mean?

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And another way to view it, you could have also written this as

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the gradient of f is equal to the partial with respect to x

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of f of x y and then all of that in the i direction.

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This is just different notation.

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Plus the partial with respect to y times f of x y.

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Not times, operated on f of x y.

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So the partial of f of x y with respect to y times

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the j unit vector.

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And because of this notation, a lot of people view this delta

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operator-- and actually it's pretty consistent when we learn

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about the divergence and the curl-- a lot of people view

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this delta operator, they actually define it.

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They say that is equal to the partial with respect to x in

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the i direction plus the partial with respect to

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y in the j direction.

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And then if you wanted to do it into three space-- that's not

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going to apply to our problem because we only care about

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two dimensions right now.

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And you'll see what I mean.

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Because even though we're applying it to a

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three-dimensional surface, but then you can do the partial

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with respect to z times the k vector.

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And then you could extend it to n dimensions, but it becomes

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very hard to visualize.

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Actually it becomes hard to visualize beyond what

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we're doing right now.

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So what does this mean?

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Well before I go into what it means, let's actually calculate

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it for this and then I'll show you what it means.

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Let's actually calculate the gradient of f.

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So the gradient of our particular function, it's a

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partial with respect to x.

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So we figure that out.

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That's 2x plus y, and then that doesn't matter anymore.

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So it's 2x plus y.

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That's the partial of this function with respect to x.

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And we multiply that in the i direction, or

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in the x direction.

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Plus the partial with respect to y, and this is 2y plus x.

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And we calculated this in the two videos on the

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partial derivatives.

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And we multiply that in the j direction.

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So what does that look like?

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Well what are each of the components of this vector?

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What is 2x plus y in the i direction?

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What does that vector look like?

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What is 2y plus x in the j direction look like?

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And this I have graphed.

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So this is the same surface we've been working with, but

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now I've plotted these points right here.

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These are just points that the software has picked to actually

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display these vectors.

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And if you look at it, this is the x-axis.

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And I can rotate this.

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I can pull it down a little bit.

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And then I can spin it around.

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I think that's pretty neat.

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But anyway.

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This parallel to this line is the vector in

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the x-axis, right?

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So at this point I've actually calculated the gradient.

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This vector says, what is the magnitude of this vector is the

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partial derivative of the function, or the partial

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derivative of the surface, or the partial derivative

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of z with respect to x.

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And it's direction, it just goes in the x direction,

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because it's that times the i unit vector.

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So this vector is the partial derivative of z with respect

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to x times the i unit vector at that point.

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So we just calculated the partial derivative

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at that point.

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It gave this length.

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And then the direction is just the i unit vector, or in the

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direction of increasing x.

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Similarly, this vector right here-- I hope you can see it.

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Let me see if I can actually zoom in a little bit.

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That might be useful.

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Zoom in, there you go.

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And I wanted to see the axes, which you can't see now that

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I'm zoomed in, but you took my word this was the x direction.

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Now this is the y direction, or the same direction

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as the j unit vector.

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The j unit vector goes in the same direction as y.

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And the magnitude is determined by the partial derivative of z

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with respect to y at that point.

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And that's the magnitude.

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And we saw there was some symmetry, so the magnitude of

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this vector is the same as the magnitude of this vector.

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And then when you add the two vectors together,

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you get this vector.

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And notice that none of these vectors have any dimensions

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in the z dimension.

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They're all kind of giving you directions in the x y plane.

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And why is that interesting?

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Well the gradient-- and this is the intuition-- the gradient

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tells you the direction in the x y plane you should travel

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in order to get the maximum slope in the z dimension.

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Or another way to view it.

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Remember the partial derivative with respect to x said what is

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the slope in the x direction.

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The partial derivative with respect to y said what is the

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slope in the y direction.

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But you could take the partial derivative with respect to any

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direction, and the gradient gives you the direction in

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which the slope is the largest.

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So let me zoom out a little bit because I want you to see the

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actual axes There's the axes So all this says if I were to

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go in this direction, I get the maximum slope.

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So if I go in this direction, my z goes up like that.

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Let me see if I can rotate this a little bit.

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I don't want to scale it anymore.

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Let me do the rotated.

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See if I can show that to you.

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So if I go in that direction in the x y plane, I get maximum z.

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If I go in that direction I get a maximum upward slope.

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That's what the gradient tells you, how do you get the

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maximum upward slope.

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And if you were to take any closed line where the z is

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constant-- well actually I don't want to get into that too

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much, but this gradient will actually be normal to any, or

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it'll be perpendicular to any curve where z is constant.

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I don't want to get too involved with that right

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now, so let's go back here.

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Maybe it's more clear if we look from below the graph.

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So if you go to this point right here, this is the

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magnitude of this vector, shows what is the slope

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in the x direction.

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The magnitude of this vector is what is the slope

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in the y direction.

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And then when you add them together, you get the gradient.

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And the gradient says, well, if I travel in this direction in

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the x y plane-- notice that none of these have any z's.

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The plane defined by all of these vectors is all

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flat in the z dimension.

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But if I would have traveled this direction in the x y

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plane, then I will get the maximum increase in z

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per unit that I travel.

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And let's actually show you what it looks like

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in the x y plane.

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So if I just go head on-- so I'm above the graph looking

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straight down at the graph, and then the colors just show kind

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of you know where we are-- if I travel in this direction in

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the x y plane, I get my maximum increase in z.

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If I'm here-- notice here the x component of the gradient is

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much larger than the y component.

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So I need to travel a little bit more in the x direction,

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and I'll get the maximum change in z if I travel there.

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Another way to think about it.

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If I'm on a hill, the gradient of that surface will tell you

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at any point what direction you need to travel in to go

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up the hill fastest.

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Or the direction which the steepness of the

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hill is maximum.

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This is more of a bowl as opposed to a hill, but anyway.

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I'll just rotate it around.

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Hopefully that makes some sense.

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And I want you to think about it a little bit more.

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And we'll do a few more problems where we just

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calculate gradients, just because I think it's useful

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to get the mechanics.

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But in my opinion at least the intuition is a little bit

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harder to get your head around than the actual mechanics,

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but once you get it, it makes a lot of sense.

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I will see you in the next video.

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