Geometric Meaning of the Gradient Vector

imagine you're climbing a mountain and you want to know which direction should you travel to for instance get up the mountain as fast as possible or alternately which directions have you travel so that you stay at exactly the same elevation that you sort of contour around the mountain without going up or down in this video we're gonna tackle that problem mathematically and then we receive it as intimately connected to the idea of the gradient vector a vector that's appeared previously in ER multivariable calculus course now I've never sort of sample Mountain here looks a little bit like a saddle point and the first thing I want to do is to put a bunch of contours on it so a contour curve is just specifying some height and then drawing a curve on that mountain which are all the points that got height so for example I might have a height where is that I said component of 2 or 1.6 or 1.7 I called level curves or just constant heights of the function now this is the three-dimensional picture but there's also a corresponding two-dimensional picture if I just take all those level curves and isolate them down in the XY plane you get a picture in the XY plane I've drawn this with perspective but maybe I'll just sort of rotate it to the standard way it's presented you have your normal XY plane and your list of contours which are really the projection of the contours down onto the XY plane now the reason the contour plot is useful is that if you identify what the Z is for each of these contours so you know the to the one point six the one point two and so forth then the way you can visualize this is saying over top of the contour associated was n equal to 2 then the function has a height of 2 and that is going to be true at any point on that particular contour or that particular level curve so thinking about this as a mountain if you're standing on some spot in the mountain you're standing on some contour the contour with a particular height and if you want to not change your height at all you just walk along that contour it actually on a map I mean this is visually obvious if I want to walk in some direction so I'm not stepping up or stepping down I'm just going for softly but then the other question is a little bit less obvious which direction should I go to maximize the change in my height if I want to climb the mountain phrasing as fast as possible so we want to try to solve that mathematically okay so let's abstract this idea a little bit let me imagine I've got some curves some R of T it's got an X component it's got a Y component so if it lives in the plane and what I'm saying is that along this curve I have some function f and then the function has a constant height anywhere along that curve that the f of the X of T Y of T is just constant okay well let's try to do a bit of calculus on that fact why don't I take for instance the derivative with respect to T of both sides of this well on the right hand side the derivative of a constant is 0 that's good easy but on the left hand side this is a composition this is the derivative of F all the x and y where x and y or themselves eats a function of T so I'm saying I want to take the derivative of a multivariable composition we've seen I take the derivative of a multivariable composition in a previous video namely it's the chain rule so if you plug in chain rule what you get is the partial of F with respect to the first variable X and then DX DT plus the partial of F with respect to the second variable Y then multiplied by dy DT and then this has to add up to zero the derivative of a constant is zero we've seen this type of computation before and indeed we can actually clean it up a little bit this is the sum of two things multiplied and so I'm gonna try to interpret this as a dot product of two vectors the first vector is the partial of f of X to X and the partial X up to Y these two components and the second vector is the derivative of X respect to T integral Y was 40 units first two components and then this is exactly the same thing because how is dot product define you multiply the first components and then you add the multiplication of the second component so indeed these are the exactly the same statements the reason why I like it written this way in terms of vectors if I can give special names to these two two vectors that I have the one on the left the parcel of effort to X and the partial of F respect to Y that vector is called the gray laughs that is our definition of the gradient of f and then the derivative of X in the variable Y with respect to T this is just well it's the derivative of R with respect to T this is the tangent vector we've also seen previously so Nicolas what I get is this expression that the gradient of F dotted with the tangent vector D R DT is equal to zero okay so let's just focus on that bottom equation the gradient of F dot the tangent vector equals zero now I have a great interpretation of what the gradient of F is they already knew what the tangent vector was that a dr/dt we've studied that previously so if you have some level curve the dr/dt is two tangent to that curve and then because the gradient of F has a dot product of zero which means it's normal what is this weighting of that it is normal to the curve at that point okay so if we return to our diagram here notice that I have a red dots in my contour plot so I'm going to pretend that I'm standing on that red dot well then I can tell you what these two different vectors are the dr/dt the tangent vector goes a long curve and if you travel in the direction the tangent vector you do not change your height you're going along that level curve so your height function is constant and then the normal vector is going to be orthogonal to the tangent vector it is pointing away from the level curve and this sort of makes sense because the level curves are gonna have different heights and so as you go from one level curves the other curve your height is changing so it leads me to think that the gradient definitely has something to do with increasing my height function but let's dive in a little bit more deeply and before we can do that I actually want to remind you of something from a previous video the idea of a directional derivative so in this graphic a lot of things going on the first thing I wanna pay attention is the arrow in the base this purple arrow this is in the doing and what it is doing is telling you the direction in which I want to compute a directional derivative and then when you decided a direction the blue curve is slicing through the surface in that particular direction if you have a direction you sort of imagine slicing it the blue curve would give you that slice and then the red line is telling you the slope of that curve in other words the slope of that red line is the directional derivative in whatever direction you're looking so this was our pictures of directional derivatives and indeed we've seen in our previous video that the so-called directional derivative in the U direction of our function f what people by the gradient dotted with the you well actually seems similar gradient dotted with you I mean that's exactly what we were just talking about before it was a gradient dotted with the tangent vector so we can sort of see there's a relationship between here but we want to be a little bit more clear what that is so let's just focus just on the equation and what I want to do is take the dot product and replace that with the length of the gradient the length of this direction vector u times cosine of theta this is just from our definition of the dot pride and I'll note here that this formula looks awfully similar to the other formula that we've just been playing around with it's a gradient dotted with something else it's just that now at the directional Druid I have a generic U versus the specific dr/dt okay so look at this gradient of that gradient of U cosine theta depending on the value of theta sometimes it's large sometimes this value is small it depends on that angle theta between the two different vectors that we did enough and the U vector well in the case where theta was PI over 2 cosine of PI over 2 is 0 so the smallest magnitude of the directional derivative ie when the slope is is smallest in magnitude happens when theta is equal to PI over 2 and if theta is PI over 2 what this means is that the gradient and the U vector have a 90 degree angle between them does the same thing is in their top part dessert so in other words the smallest magnitude of the directional derivative occurs when you have a dr/dt as your direction so if you're traveling in the direction dr/dt you're traveling along that level curve you're going tangent to that level curve then as you'd expect the directional derivative is just zero because if it brought a lot of level curve you're not changing your function at all your slope is just zero to go along a level curve and so everything just makes sense here our directional derivative traveling in the tangential direction is zero as you expect okay what about the largest magnitude well cosine the maximum value of that is one and one of the cases that that happens is when theta is equal to zero so I can say that the largest magnitude is gonna happen when theta is equal to zero of course they was equal to pi you get the same value but the negative N instead so I don't care about pus or - then unless you say be equal to zero okay so in this scenario I have my gradient vector I have my u vector and I'm saying that there is in fact an angle of nothing in between them this is just saying that the U vector is parallel to the gradient vector and so now we have our interpretation of the gradient vector gradient of F is the direction you travel it's placing the role of the you it's the direction you travel when you want to increase your slope as much as possible and what to have the largest directional derivative so the way I like to think about this is that you can have your directional derivatives pointed in any different directions but there's two that are special when you go tangent to the curve that is going to be your minimal slope and when you go normal to the curve that's gonna be your maximum slope all right so let's return to our contour plot let's put our point on it once again and try to put these two different vectors so first we have the tangent vector of the dr/dt ND this goes along that level curve and so we're gonna have no increase in slope a KH direction over to the zero and then we're gonna have our normal vector and our normal vector is going to point orthogonal to the curve or normal to the curve and it sort of makes sense because when you go in that direction of the gradient of F it's pointing towards another level curve you're at one level curve it points to a second to a third and so forth it makes sense that the normal vector would be consistent with the idea that's the direction you go when you want to increase your height as fast as possible it's the maximal directional derivatives okay so the next thing I want to do is try to interpret this in a little bit more of a real-world context this is a topographical map it shows the mountain called the golden find the Golden Hind by the way is the tallest mountain in the mountainous island chain of Vancouver Island which is where I live I really love this particular mountain it looks a little bit like this in the winter it's kind of a big and imposing mountain I do want to note that now do you want to note that the mountain itself has this really big imposing front cliff face but that then in front of the mountain they sort of a sloping hills sort of relatively flat at least compared to the steepness of the mountain itself in the front of it ok so let's go back to the top of graphical map know what's happening here is what you're seeing is a whole bunch of about level curves of contours so for example a contour like this one that's got 21 20 meters on it just means that that's about a little over 2,000 meters tall when you're walking along that contour by the way you'll notice that there's this initial region where the contours are all really close together what that means is that with small horizontal changes you get big increases in height that was those big cliffs we just saw and then there's this other sort of larger portion where the contours are further apart that means over a larger horizontal distance are you going to be able to get to an increase in height to go from one contour to the next those are sort of those flatter more rolling hills we saw it in Punta mountain now if you don't want to climb the mountain well you don't really wonder about the cliff face and Leicester more skill than I am when I just go up the backside here kind of up this little Ridge and this Ridge is nice because as you go along and it's not as steep as going up sort of the big cliff it's going to go around the back from the picture I took and showing you earlier ok ok so let's get back to the calculus then if I want to look at some point so I just put a random dog on I want to imagine you're standing there I'm on one of those corners I'm on one of those level curves then the way we can interpret this is that the tangent vector is pointing along that curve and this is the dr/dt and then the normal vector is going to point normal to that and well it's going to point exactly up the cliff face ok and then at the same point when we actually look at the image of it well I put my point on the mountain and then if I look at the tangential vector this would just be going along at the same height just move it along the mountain but not went up not going down and then if you go in the direction of the gradient vector you would rise as fast as possible you're going directly at these this now I should just clarify that the gradient vector leads down in the XY plane it's truly a vector that's pointing into the mountain from our perspective not going up in the Z component but I just drew it this way to try to give you a little bit of a perspective on it but but truly it should be thought of as going into the mountain living in the XY plane know when you're actually climbing mountains and you're trying to use a topographical map to orient yourself and to plan or route through the mountains the analysis of these contours are actually really really important so they're really sort of winter sport involving skiing or snowshoes where you don't want to do big Rises and then drops in elevation as much as you can avoid excessive you don't respond to work in any plot that you want to do often you want to try to contour or to stay as close to a contour as possible so you can sort of go horizontally without the much larger amount of work of trying to climb them out and then go back down again and climb it back down again you sort of want to contour around and so this is a very common thing when you climb and try to sort of figure out that the minimal pathway to reduce the amount of uphill but you actually have to do is you're navigating alright I hope you enjoyed that video I tried a little bit of fun with this particular one blennies in calculus with some actual mountain climbing ideas if you enjoyed the video then please give a like for you to gather any questions about the video leave them down in the comments below and we'll do some more math in the next video

Channel: Dr. Trefor Bazett

Views: 174,995

Keywords: Math, Solution, Example, Gradient, Gradient Vector, Directional Derivative, Partial Derivative, Multivariable, Calculus, Calc III, direction, maximum increase, contour, level curve, geometric interpretation, slope, gradient and graphs, input space

Id: QQPz3eXXgQI

Length: 14min 51sec (891 seconds)

Published: Sun Jun 21 2020