Gradient 1 Introduction to the gradient
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- Let's learn a little bit about the gradient, and we'll use the
- same function we've been using since we're pretty familiar
- with it's graph and it's partial derivatives.
- So let's say that f of x y is equal to x squared plus x
- times y plus y squared.
- Now we're going to take the gradient of this, and then I'll
- give you a little bit of intuition on what
- a gradient is.
- We're going to take the gradient in two-dimensional
- space, and you'll see what that means in a second.
- You can extend this to any dimension.
- So the gradient that we use this vector differential
- operator-- this upside down triangle-- the gradient of f is
- equal to the partial of f in the x direction times the unit
- vector i-- which is the unit vector in the x direction so it
- gives the magnitude of the slope and the x direction, and
- then it points the vector in that direction-- plus the
- partial with respect to y in the j direction.
- And obviously if there's a function of more variables you
- could keep going and you'd multiply the magnitude of the
- derivative in each of those dimension's directions times
- the unit vector in that dimension.
- Well what does this mean?
- And another way to view it, you could have also written this as
- the gradient of f is equal to the partial with respect to x
- of f of x y and then all of that in the i direction.
- This is just different notation.
- Plus the partial with respect to y times f of x y.
- Not times, operated on f of x y.
- So the partial of f of x y with respect to y times
- the j unit vector.
- And because of this notation, a lot of people view this delta
- operator-- and actually it's pretty consistent when we learn
- about the divergence and the curl-- a lot of people view
- this delta operator, they actually define it.
- They say that is equal to the partial with respect to x in
- the i direction plus the partial with respect to
- y in the j direction.
- And then if you wanted to do it into three space-- that's not
- going to apply to our problem because we only care about
- two dimensions right now.
- And you'll see what I mean.
- Because even though we're applying it to a
- three-dimensional surface, but then you can do the partial
- with respect to z times the k vector.
- And then you could extend it to n dimensions, but it becomes
- very hard to visualize.
- Actually it becomes hard to visualize beyond what
- we're doing right now.
- So what does this mean?
- Well before I go into what it means, let's actually calculate
- it for this and then I'll show you what it means.
- Let's actually calculate the gradient of f.
- So the gradient of our particular function, it's a
- partial with respect to x.
- So we figure that out.
- That's 2x plus y, and then that doesn't matter anymore.
- So it's 2x plus y.
- That's the partial of this function with respect to x.
- And we multiply that in the i direction, or
- in the x direction.
- Plus the partial with respect to y, and this is 2y plus x.
- And we calculated this in the two videos on the
- partial derivatives.
- And we multiply that in the j direction.
- So what does that look like?
- Well what are each of the components of this vector?
- What is 2x plus y in the i direction?
- What does that vector look like?
- What is 2y plus x in the j direction look like?
- And this I have graphed.
- So this is the same surface we've been working with, but
- now I've plotted these points right here.
- These are just points that the software has picked to actually
- display these vectors.
- And if you look at it, this is the x-axis.
- And I can rotate this.
- I can pull it down a little bit.
- And then I can spin it around.
- I think that's pretty neat.
- But anyway.
- This parallel to this line is the vector in
- the x-axis, right?
- So at this point I've actually calculated the gradient.
- This vector says, what is the magnitude of this vector is the
- partial derivative of the function, or the partial
- derivative of the surface, or the partial derivative
- of z with respect to x.
- And it's direction, it just goes in the x direction,
- because it's that times the i unit vector.
- So this vector is the partial derivative of z with respect
- to x times the i unit vector at that point.
- So we just calculated the partial derivative
- at that point.
- It gave this length.
- And then the direction is just the i unit vector, or in the
- direction of increasing x.
- Similarly, this vector right here-- I hope you can see it.
- Let me see if I can actually zoom in a little bit.
- That might be useful.
- Zoom in, there you go.
- And I wanted to see the axes, which you can't see now that
- I'm zoomed in, but you took my word this was the x direction.
- Now this is the y direction, or the same direction
- as the j unit vector.
- The j unit vector goes in the same direction as y.
- And the magnitude is determined by the partial derivative of z
- with respect to y at that point.
- And that's the magnitude.
- And we saw there was some symmetry, so the magnitude of
- this vector is the same as the magnitude of this vector.
- And then when you add the two vectors together,
- you get this vector.
- And notice that none of these vectors have any dimensions
- in the z dimension.
- They're all kind of giving you directions in the x y plane.
- And why is that interesting?
- Well the gradient-- and this is the intuition-- the gradient
- tells you the direction in the x y plane you should travel
- in order to get the maximum slope in the z dimension.
- Or another way to view it.
- Remember the partial derivative with respect to x said what is
- the slope in the x direction.
- The partial derivative with respect to y said what is the
- slope in the y direction.
- But you could take the partial derivative with respect to any
- direction, and the gradient gives you the direction in
- which the slope is the largest.
- So let me zoom out a little bit because I want you to see the
- actual axes There's the axes So all this says if I were to
- go in this direction, I get the maximum slope.
- So if I go in this direction, my z goes up like that.
- Let me see if I can rotate this a little bit.
- I don't want to scale it anymore.
- Let me do the rotated.
- See if I can show that to you.
- So if I go in that direction in the x y plane, I get maximum z.
- If I go in that direction I get a maximum upward slope.
- That's what the gradient tells you, how do you get the
- maximum upward slope.
- And if you were to take any closed line where the z is
- constant-- well actually I don't want to get into that too
- much, but this gradient will actually be normal to any, or
- it'll be perpendicular to any curve where z is constant.
- I don't want to get too involved with that right
- now, so let's go back here.
- Maybe it's more clear if we look from below the graph.
- So if you go to this point right here, this is the
- magnitude of this vector, shows what is the slope
- in the x direction.
- The magnitude of this vector is what is the slope
- in the y direction.
- And then when you add them together, you get the gradient.
- And the gradient says, well, if I travel in this direction in
- the x y plane-- notice that none of these have any z's.
- The plane defined by all of these vectors is all
- flat in the z dimension.
- But if I would have traveled this direction in the x y
- plane, then I will get the maximum increase in z
- per unit that I travel.
- And let's actually show you what it looks like
- in the x y plane.
- So if I just go head on-- so I'm above the graph looking
- straight down at the graph, and then the colors just show kind
- of you know where we are-- if I travel in this direction in
- the x y plane, I get my maximum increase in z.
- If I'm here-- notice here the x component of the gradient is
- much larger than the y component.
- So I need to travel a little bit more in the x direction,
- and I'll get the maximum change in z if I travel there.
- Another way to think about it.
- If I'm on a hill, the gradient of that surface will tell you
- at any point what direction you need to travel in to go
- up the hill fastest.
- Or the direction which the steepness of the
- hill is maximum.
- This is more of a bowl as opposed to a hill, but anyway.
- I'll just rotate it around.
- Hopefully that makes some sense.
- And I want you to think about it a little bit more.
- And we'll do a few more problems where we just
- calculate gradients, just because I think it's useful
- to get the mechanics.
- But in my opinion at least the intuition is a little bit
- harder to get your head around than the actual mechanics,
- but once you get it, it makes a lot of sense.
- I will see you in the next video.
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