Change of Variables & The Jacobian | Multi-variable Integration

whenever we want to integrate a function over a region we need to decide what coordinate system should we use for instance just a rectangle is really easy to compute in Cartesian or rectangular coordinates well a circle is very easy to compute in polar coordinates but not so easy in rectangular coordinates however we can move on to any coordinate system that we wish to define and if that coordinate system makes it easy to express reason that it might be useful using that coordinate system consider for example this region there's two horizontal lines y 2 1 y 2 and that's sort of the same kind of thing we've seen in cartesian where you have specify different Y values but if I look at the two red lines the line y with X minus 1 and X minus 2 there's a certain symmetry on the two different sides here but it's it's not vertical line symmetry as is often represented in Cartesian coordinates so what if I do a change of variables that is what if I introduced new variables U and V for the V is just gonna be equal to the Y values that I said I was happy with those horizontal lines those are perfectly fine but for the two diagonal lines that are not easily expressed by X equal to a fixed value if I set u equal to X minus y then the one line is just u even one and the other line is u equal to 2 these are just constants indeed if I try to draw the exact same region but now in the UV plane that is expressing it in terms of U and V opposed to x and y what I get is just a rectangle it's just the rectangle where the V values are between 1/2 and the u values are between 1 & 2 so my plan is that this is the me way easier to integrate in this coordinate system because my limits of integration is going to be 1 & 2 for you and 1 & 2 for V so how do I actually integrate in some new a coordinate system let's think back to what we used to do for single variable u substitution the idea was the following suppose I have an integral of some function of X DX then what I can do is take X and write it as some function of a new variable a new variable U and then what the U substitution did was it allows us to rewrite this this new integral is an integral in terms of U the function of X has now turned into F of G of U because X is defined to be G of U and the DX transforms into a tu but there's an important point to note there is also this new factor this scaling factor this G prime of U and that was u substitution well note that this formula may seem exactly backwards to how you recall it in single variable calculus typically in single variable calculus you would actually set u to be a function of X opposed to X being a function of U it doesn't matter which way you write it which you think of a sort of the new variable and the old variable but I'm going to do it this way because I'm about to have a human of V I want to think of the new V is converting the messy XY into a nice you would be according Center so I like this order of presenting it nevertheless let's go and look at what it's going to be in multivariable calculus well you begin sort of much the same way you have an integral in terms of X and Y and it's integrated over some ticular region and then you say I need to take the X and write that in terms of U with v some function G of U with u I'm gonna take the Lion write that in terms of UV some function H of U would be then if I plug this all in I want to get an interval with respect to U and V well the function evaluates now in terms of U and V the limits of integration are no longer the region now we call it the region G that's what happens when you transform under these equations and then the DX dy transforms into a dudv but just like in the single variable case you have to have a multiplicative factor we're going to call this multiplicative factor no longer G Prime we call it J of U and V here J is called J for the Jacobian at the end of the video I'm going to talk about the intuition of how to come up with this J of U and V but for now I'm just going to state it je u and V the Jacobian is just the determinant of particular matrix a matrix of partial derivatives and the idea here is that this represents the scaling factor from when you shift your areas thought of in terms of the XY coordinate system to being areas in the UV coordinate system you need to get a scaling factor and this je gives you that scaling factor all right so let's look at the example that we had before where are you as X minus y and our V was just equal to Y and then if I take that exact same region I now represent it in the UV coordinate system it looks a lot nicer first thing I want to do is try to write it in the way that's can paddle of my formulas where I had X as a function of U and V and Y is a function of U and V so I'm just going to take these formulas I'm just gonna rewrite them in that way and if I do that of course Y is equal to V is the same thing as V Y I plug that in and I get X is equal to u plus B alright so let's compute out that Jacobian this determinant well most of these parcels are straightforward what's the partial of X back to you well it's just 1 and so on so if we compute what these are going to be it's just the determinant of the matrix 1 1 0 1 and that's just equal to the value of 1 so in this case the Jacobian is just x 1 so here that is my final formula the double integral over the original region the sort of parallelogram one if I take that region and integrate some function of x and y over it's the same thing as the much simpler region the region in the UV coordinate system where the you in ruble this goes between one and two and the interval suppose between one and two the function whatever it is I haven't told you would be represented down when in terms of U and V u multiplied by that one for the Jacobian and then you get your result have you integrate respect to U and V you might want to do this transformation two variables for multiple reasons it might be that your region is really messy and you can transform it into a much nicer region with a substitution or it might be that your integrand is really messing you don't have any degree back and then with the substitution you can now integrate the integrand so there's multiple reasons why you might wish to do a change of variables final thing i want to talk about in this video is motivate the J of U and V the Jacobian why was that the case so don't imagine that I'm in the UV coordinate system and I'm going to transform a little region a little area in the UV coordinate system into the XY coordinate system that is imagine I just have a little rectangle this is a rectangle and it's can be described by having two different vectors one vector is a little change in U and no change in V and the other one consists of no change in U and a little change in V when I integrate of course I take a region I think of breaking it up into a bunch of little regions so in the UV coordinate system these little regions are simple they're just little rectangles but let's see what this little region looks like all the way back in the XY coordinate system indeed this green arrow goes some where the red arrow goes somewhere and whatever you get defines a parallelogram as in the original square defined by these two vectors transforms into a parallelogram defined by wherever those two vectors and not but where do they end up well if you think of the green vector as the result of just first a small nudging you and then you transform it into x and y then the result you get will what change in X do you get its the resulting change from nudging the you a little bit in other words what you get is a change in X with respect to u times that little distance bu and likewise for y it's the partial of Y with respect to u times to u so these are the little nudges in the X in the y direction that result from nudging you a little bit then likewise for the red vector you get a similar kind of construct they nudge in the X Direction is just the change in X with respect to V times this DV the change in the Y Direction is the change in Y with respect to this V DV now back in the original UV coordinate system the area was easy to express it was just D u times DV so if your big region you make it a bunch of little rectangles these little rectangles have areas to u DV exactly what we've seen when we defined double integration but now if I ask the area for this same region but then map back into the XY important system but area of the parallelogram is a cross product so what is the area of this new parallelogram well it's just going to be the cross product of these two little vectors and the cross product those two different vectors is the determinant that we've seen before indeed this is nothing but the Jacobian so what does the Jacobian represent it says you take a region in UV a little area UV you transform it into the XY coordinate system well you get a multiplicative factor on there areas the dudv transforms to this matrix times a dudv so you're going to integrate with respect to these new coordinate systems that's fine that's great it's easy but you have to bring along this multiplicative factor this J of U and V if you have a question about this video leave it down in the comments below real mathematicians here we appreciate algorithms so let's just hope the you to bail you out by and giving this video a like and finally if you want to watch more multivariable calculus videos this video is part of a larger playlist a multivariable calculus so you can check out those videos here and we'll more math in the next video

Channel: Dr. Trefor Bazett

Views: 53,805

Rating: 4.9597197 out of 5

Keywords: Math, Solution, Example, Jacobian, Change of Variables, u-subs, integration, calculus, multivariable, proof, derivation, geometric meaning

Id: wUF-lyyWpUc

Length: 10min 6sec (606 seconds)

Published: Wed Dec 18 2019