![]() ![]() Because if you think about these as lines, And the more you zoom in, the more they pretty much It's gonna be the one that connects them pretty much perpendicular The shortest distance? And if you think of them asīeing roughly parallel lines, it shouldn't be hard to convince yourself that the shortest distance isn't gonna be, you know, any of those. Thinking of the fastest as constant-length vectors, what increases it the most, we'll be thinking, constant You know, which one does it the fastest? And this time, instead of You know, you're looking at all of the possible different So another way we can thinkĪbout the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1. Because if you change the output by just a little bit, the set of in points that look like it is pretty much the same but And if I were a better artist, and this was more representative, it would look like a line that's parallel to the original one. That represents, you know,Īnother value that's very close. All right, so let's say you're taking a look at a contour line, another contour line. These different directions and say which one increases x the most? But another way of doing it would be to get rid of them all and just take a lookĪt another contour line that represents a slight increase. So if you imagine all the possible vectors kind of pointing away from this point, the question is, whichĭirection should you move to increase the value of f the fastest? And there's two ways Video about how to interpret the gradient in the context of a graph, I said it points in theĭirection of steepest descent. But the more you zoom in, the more it looks like a straight line. And, you know, it might notīe a perfect straight line. Going on in that region? So you've got some kind of contour line. We'll take that guy and kind of imagine zooming in and saying what's Clear up all of the information about it. You know, you go down here, this vector's perpendicular Of the given points around, if the vector is crossing a contour line, it's perpendicular to that contour line. And remember, we scaledĭown all the vectors. So I'll go ahead andĮrase what I had going on. But it's probably gonna be scaled down because of the way we See the vector that has an x component of oneĪnd a y component of two. So we're looking somehow toĭraw the vector one, two. And at this point, the point is two, one. You would plug in the vector and see what should be output. So that would be xĮquals two, y equals one. You know, at every given point, xy, so you kind of go like x equals two, y equals one, let's say. And this can be visualizedĪs a vector field in the xy plane as well. And then kind of the reverse for when you take the partial derivative The derivative of this whole thing is just equal to that constant, y. This for our function, we take the partialĭerivative with respect to x. And the second component is the partial derivative And it's a vector-valued function whose first coordinate is the partial derivative Our little del symbol, is a function of x and y. And the gradient, if you'll remember, is just a vector full of the So all of these lines, they're representing constant Is x times y equal to two? And that's kind of like the graph y equals two over x. And a way you could think about that for this specific function is you saying hey, when So you might be thinking that you have, you know, let's say you want a the constant value for f of And the contour map for x times y looks something like this. ![]() And I have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable. ![]() All right, so this right here represents x values. So what I'm gonna do is I'm gonna go over here. This with a contour map just on the xy plane. So here I want to talk about the gradient and theĬontext of a contour map. ![]()
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |