Khan Academy Video 2 (Gradient vs. Directional Derivative) #khanacademytalentsearch

Rhonda I'm back with my second Khan Academy video recall that in the first one we defined the gradient of a function of two variables and the directional derivative of a function in the direction of a unit vector U now in this video I want to try and convince you that some of the important properties of the gradient are actually derived from features of the directional derivative so in order to see that we have to take a side trip for a moment and recall an alternate definition of the dot product of two vectors so that should be a straight vector and let's let theta be the angle between those two vectors while the alternate definition is that the dot product of U and V is the length of U times the length of V times the cosine of the angle between them this definition is extremely useful actually this definition is usually given in terms of coordinates the product of the first coordinates plus the product of the second coordinates but this is particularly useful because when two vectors are perpendicular or orthogonal then the angle between them is 90 degrees so the cosine is zero so in the case of perpendicular vectors the dot product is zero and it goes the other way if the dot product of two vectors is zero don't we know that they must be perpendicular that's an extremely useful feature of the dot product so now let's just return to the directional derivative which is defined as a dot product and apply that definition and see what it tells us well this is the length of the gradient times the length of U times the cosine of the angle between those two vectors so let's say this is you and this is the gradient and Stata well we know that use a unit vector so this is just one so this is actually equal to the length of the gradient vector times the cosine of the angle between you and the gradient now we know something important about the cosine namely it's always between 1 and minus 1 and that tells us something fairly interesting about the directional derivative it tells us what is the largest value the directional derivative can have and what is the smallest value the directional derivative can have so let's take a look at that dirt when Coast theta is 1 the directional derivative is a max because 1 is the largest value that this can have and what is that value it's just the value of the length of the gradient by contrast when cos theta is minus 1 the directional derivative is a minimum because that's the smallest otherwise this is a number between minus 1 and 1 it's a minimum and that value this would now be minus 1 this is the negative of the length of the gradient vector so we want to keep that in mind and look at the surface that's going to make this a little more transparent well I want a surface that looks a little bit like a mountain because we're gonna be doing some skiing shortly so let's go back to our three and let's now take a function maybe a paraboloid let's go hear something like that so now we have a function of two variables let's take a point down here let's take a point X naught Y naught and plot that point on our surface okay now it is a natural question to ask in which direction should we go so that the rate of change along this curve is going to be the greatest or the least set another way what are the directions of maximum descent and ascent okay well we know that the direction of maximum ascent is going to be in the direction of the gradient in other words when you and the gradient are in the same direction now it's the same not same the direction of maximum descent is in the direction of minus the gradient but let's see what that looks like in this picture okay one thing that I have not told you it's a property of the gradient vector that's extremely important but due to budgetary constraints we did not have time to prove this is we've actually done most of the work but at any rate if we slice through a surface with a plane where Z is a constant then what do we get well in this case it looks like we get a circle or an ellipse depending on what the equation of that surface is and it is a fact that so this is called a level curve and it is a fact and an important fact that at any point the gradient vector is always perpendicular to level curves so the gradient vector is perpendicular to level curves what does that mean well remember the gradient vector is a 2-dimensional vector not a three-dimensional vector so it means that once we have this curve that is kind of a tangent line to that curve and perpendicular to that in one of those two directions will be the gradient vector now I drew it so it kind of looks like it's pointing upwards but it's down here we could draw it in the XY plane so let's think about this in terms of doing some scheme now I have a family friend Katja she is in college now but I have discussed mathematics with Katja since she was in eighth grade and she's a very bright young woman let's see Koch and there she is she's also quite fashionable so we'll give her some ski gear she has lived all over the world and she has in particular skied the Matterhorn let's make her skis vectors why not give her some poles now CACCI has skied the Matterhorn which is in the Alps it's a mountain between Switzerland and Italy and it looks a little bit like that so let's assume that Katja is at this point on the surface of the Matterhorn now because I care about her safety and well-being when kaca decides to ski downhill I would of course hope that she does well most reasonably cautious skiers would do and that is zigzag down the hill or down the mountain however let's assume that for a moment cocktail wants to go down that and experience the maximum rate of descent so let's say she wants to go in the direction that will give her the maximum rate of descent well we know that that is - the direction of the gradient now in this case because of what we said about gradients being perpendicular to level curves the gradient is actually going to be this vector pointing inward why is that because when we take this vector and plot the corresponding points along the surface we get a curve and so it's that curve that has the maximum rate of change in that direction what if we go so this in fact is the gradient vector what if we go in this direction - the gradient vector well what does that mean it means we take these points and we plot them on the surface and if Koch's fees this way she will enjoy the maximum rate of descent down the Matterhorn now there's something slightly subtle here because she's going up or down the hill down probably it feels like maybe the vector should point down like this or something like that but remember the gradients a - it doesn't because if she went in that direction she would go into the ground if she went in this in the direction like this she would fly off into space which we certainly don't want to happen to Katja but the vector that we're interested in the gradient is a 2-dimensional vector so all it means is that it creates a curve along the surface and it's that curve that will have the maximum or minimum rate of change now this property of the gradient is extremely important if you've ever seen water trickling down a mountainside from a stream perhaps it always takes the path of maximum descent that means that at any given point the water will travel in the direction of the gradient and many of the fundamental applications of the gradient depend on those properties so I hope that I have shown you in the second video at least to some extent that the directional derivative really does contribute to some of the success that the gradient enjoys I don't really have time to tell you more there is always more but that can be for another time thank you so much for your attention

Channel: Rhonda Hughes

Views: 40,691

Rating: 4.9600501 out of 5

Keywords: #khanacademytalentsearch, Gradient, Directional Derivative, multivariable calculus, Partial Derivatives, Vectors

Id: 3xVMVT-2_t4

Length: 11min 21sec (681 seconds)

Published: Mon Jun 22 2015