Understanding the Jacobian Determinant and Its Geometric Meaning | Multivariable Calculus

[00:00:00]in this video i want to talk about something called the jacobian determinant and it's more or less just what it sounds like it's the determinant of the jacobian matrix that i've been talking to you the last couple videos about uh and before we jump into it i just want to give a quick review of how you think about the determinant itself just in an ordinary linear algebra context so if i'm taking the determinant of some kind of matrix let's say 3 0 1 2 something like this to compute the determinant you take these diagonal terms here so you take 3 multiplied by that 2 and then you subtract off the other diagonal subtract off 1 multiplied by 0. and in this case that evaluates to 6.

[00:00:46]but there is of course much more than just a computation going on here there's a really nice geometric intuition namely if we think of this matrix 3012 as a linear transformation as something that's going to take this first basis vector over to the coordinates 3 0 and that second basis vector over to the coordinates 1 2 you know thinking about the columns you can think of the determinant as measuring how much this transformation stretches or squishes space and in particular you'll notice how i have this yellow region highlighted and this region starts off as the unit square a square with side lengths one so it's area is one and there's nothing special about this particular region it's just nice as a canonical shape with an area of one so that we can compare it to what happens after the transformation ask how much does that area get stretched out and the answer is it gets stretched out by a factor of the determinant that's kind of what the determinant means is that all areas if you were to draw any kind of shape not just that one square are going to get stretched out by a factor of six and we can actually verify looking at this parallelogram that the square turned into it has a base of three and then the height is two and three times two is six and that has everything to do with the fact that this three showed up here and this two showed up there so now let's think about what this might mean in the context of what i've been describing in the last couple videos and if you'll remember we had a multivariable function something that you can write out as f1 with two inputs and then the second component f2 also with two inputs and the function that i was looking at that we were kind of analyzing to learn about the jacobian had the first component x plus sine of y x plus sine y and the second component was y plus the sine of x and the idea was that this function is not at all linear it's going to make everything very curvy and complicated however if we zoom in around a particular region which is what this outer yellow box represents zooming in it will look like a linear transformation in fact i can kind of play this forward and we see that even though everything is crazy inside that zoomed in version things loosely look like a linear function and you'll notice i have this inner yellow box highlighted and this yellow box inside corresponds to the unit square that i was showing in the last animation and again it's just a placeholder as something to watch to see how much the area of any kind of blob in that region gets stretched so in this particular case when you play out the animation areas don't really change that much they get stretched out a little bit but it's not that dramatic so if we know the matrix that describes the transformation that this looks like zoomed in the determinant of that matrix will tell us the factor by which areas tend to get stretched out and in particular you can think of this little yellow box and the factor by which it gets stretched and as a reminder the matrix describing that zoomed in transformation is the jacobian it is this thing that kind of holds all of the partial differential information you take the partial derivative of f with respect to x sorry partial of f 1 of that first component and then the partial derivative of the second component with respect to x and then on the other column we have the partial derivative of that first component with respect to y and the partial derivative of that second component with respect to y and if you let's see and we'll close this off close off this matrix and if you evaluate each one of these partial derivatives at a particular point at whatever whatever point we happen to zoom in on in this case it was negative 2 1 once you plug that into all of these you get some matrix that's just full of numbers and what turns out to be a very useful thing later on in multi-variable calc concepts is to take the determinant of that matrix to kind of analyze how much space is getting stretched or squished in that region so in the last video we worked this out for the specific example here where that top left function turned out just to be the constant function 1 right because we were taking the partial derivative of this guy with respect to x and that was 1. and likewise in the bottom bottom right that was also a constant function of 1.

[00:05:02]and then the others were cosine functions this one was cosine x because we were taking the partial derivative of this second component here with respect to x and then the top right of our matrix was cosine of y and these are in general functions of x and y because you know you're going to plug in whatever the input point you're zooming in on and when we're thinking about the determinant here let's just go ahead and take the determinant in this form in the form as a function so i'm going to ask about the determinant of this matrix or maybe you think of it as a matrix-valued function and in this case we do the same thing i mean procedurally you know how to take a determinant we take these diagonals so that's just going to be 1 times 1 and then we subtract off the product of the other diagonal subtract off cosine of x multiplied by cosine of y and as an example let's plug in this point here that we're zooming in on negative 2 1. so i'm going to plug in x is equal to negative 2 and y is equal to 1. and when you plug in cosine of negative 2 that's going to come out to be approximately negative 0.42 and when you plug in cosine of y cosine of 1 in this case that's going to come out to be about 0.54 and when we multiply those when we take 1 minus the product of those it's going to be about negative 0.227 and that's all stuff that you can plug into your calculator if you want and what that means is that the total determinant evaluated at that point the jacobian determinant at the point negative 2 1 is about 1.0 sorry 1.227 so that's telling you that areas tend to get stretched out by this factor around that point and that kind of lines up with what we see we see that areas get stretched out maybe a little bit but not that much right it's only by a factor of about 1.2 and now let's contrast this if instead we zoom in at the point where x is equal to zero and y is equal to one so i'm going to go over here and all i'm going to change all i'm going to change that x is equal to 0 and y will still equal 1.

[00:07:19]and what that means is that cosine of x instead of being negative 0.42 well what's cosine of 0 that's actually precisely equal to 1 right we don't have to approximate on this one which means when we multiply them 1 times 0.54 well that that's going to now be about 0.54 right so this one once we actually perform the subtraction instead when you take 1 minus 0.54 that's going to give us 0.46 so even before watching because this determinant of the jacobian around the point 0 1 is less than 1 this is telling us we should expect areas to get squashed down precisely they should be squashed by a factor of 0.46 and let's see if this looks right right we're looking at the zoomed in version around that point and areas should tend to contract around that and indeed they do you see it got squished down it looks like by a fair bit and from our calculation we can conclude that they got scaled down precisely by a factor of 0.46 that's what the determinant means so like i said this is actually a very nice notion throughout multivariable calculus is that you look at a tiny little local neighborhood around a point and if you just want to get a general feel for does this function as a transformation tend to stretch out that region or to squish it together you know how much do areas change in that little neighborhood that's exactly what this jacobian determinant is you know built to solve so with that i'll see you guys next video