Alternative view of integration | Differential forms #2

The integral of f ofx dx from a to b is usually interpreted as the area under a graph. But there is an alternative view offered by differential forms which I would call measurement fields. The idea is to stay in one dimension and focus on the interval between a and b. Our goal is to make its quote unquote length equal to the integral. We can do this using our imaginary ruler. Initially, just like a normal ruler, it has markings at regular intervals. When we measure the length of an object, what we really mean is look at how many markings this object has spanned across. In this case, the number is six. So the length is six. But now let's change this ruler up a bit. Stretch and squish the spaces between the tick marks so that they appear at irregular intervals. If we still define quote unquote length as how many markings this object spans, then the length is now measured to be 16. By changing the tick marks on the ruler, we can make this length whatever we like. In particular, there should be a way to make this exactly equal to the integral we started off with. The trick is to make the density of the markings equal to f ofx at the point x. We'll go through this with an example. Let's say this object goes from 0 to 6. So a normal ruler would have looked like this and this object would have normally spanned across six markings. Now say f ofx equals 2 in the first half of the object and 0. 5 in the second half. To make the length equal to the integral, we double the density of the tick marks in the first half. So this part now spans six of them instead of just three. Twice the amount as before. Similarly on the other half where f ofx equals 0. 5 we half the density so that this part spans one and a half tick marks instead of the original three. Half the amount as before. At the end, we can count the number of markings spanned by the entire object to be 7 and 1/2. So the quote unquote length is 7 and 1/2. You can check that the integral in this case does equal 7 and 1/2 by separating it into two parts. The first part integrates the function which equals 2 here from 0 to 3 and the second part integrates the function which equals 0. 5 here from 3 to 6. A bit of computations will show that this integral evaluates to 7 and 1/2 exactly the same as what we have obtained by changing the roller tick marks. This works because the tick marks being f ofx times a dense means that things will span across f ofx times as many tick marks. If its normal length is delta x, then its new length will be measured as f ofx delta x. When we aggregate the contributions from these little sections, the total length is the integral. This weird ruler is essentially the differential form f ofx dx and it provides an alternative view of integration. The integral is the number of markings spanned by the interval AB according to this ruler. Before we generalize this weird ruler to higher dimensions, there are two wrinkles to iron out. One, what if f ofx is negative somewhere? What does a negative density mean? A quick fix is to separate the tick marks into positive markings colored in green and negative markings colored in red. when f ofx reaches negative values. So when we measure this interval, we have a positive count when it reaches a green marking, but a negative red one. In this

illustration, the quote unquote length is -2 because of one positive marking and three negative ones. Another thing is how to encode orientations. This refers to the fact that the integral from A to B is the negative of that from B to A. A quick fix is to indicate with a little arrow for each tick mark. If the direction you integrate matches the direction on the tick marks, then proceed as you normally do. But if the direction is against those on the tick marks, you gain a minus sign. Just to drive home this intuition before going to higher dimensions, I want to show you how the differential form x dx looks like on the ruler. It looks like this. You'll notice that since the density of the markings is x, they go denser as you move away from zero because the magnitude of x increases. The difference is that on the positive side the tick marks are positive colored in green but on the negative side they are negative colored in red. For consistency with later parts of the video, we reduce each tick mark to a dot. With that, let's generalize to a two-dimensional integral where we integrate a function over a two-dimensional region D. The philosophy is very similar, but this time we make the quote unquote area equal to the integral. Instead of counting tick marks on a ruler, this time we count the number of boxes this region covers. Just like what you have probably done in school. Similar to the one-dimensional case, we want to change the density of the boxes. For example, if the value of f at this point xy is four, then we make the boxes four times as dense here. So that this covers four times as many boxes here. After changing the densities of the boxes throughout the plane according to the values of f, we obtain an irregular grid pattern. Then the integral is the quote unquote area of this region as measured by counting the number of boxes it covers. While this is a perfectly valid way to visualize f ofxyd and in fact some do use this for such a differential form. We will modify this visualization a tiny bit. We replace each grid with a dot in the middle so that now the quote unquote area is measured by the number of dots inside the region instead of boxes. And similar to the previous one-dimensional case, we can distinguish between positive dots colored in green and negative dots colored in red depending on whether f is positive or negative there. However, unlike the 1D case, the notion of orientation is a lot more subtle. If we focus on one dot, we indicate its orientation by two arrows and they are ordered. In this case, the right arrow is first and the up arrow is second. So the second direction is to the left of the first. This orientation roughly refers to that in f ofxyd. dx is kind of like the first direction and dy is kind of the second direction. That means f ofxy dydx has the opposite orientation where the second direction is to the right of the first. So we actually have dx dy not equal to dydx. And in fact, dx dy is negative dydx. This tells us that we shouldn't treat these things as ordinary numbers. And nowadays, mathematicians use a symbol known as the wedge product to indicate it isn't multiplication in the usual

sense. More about it in a future video. Now in the overall picture we expect to see many dots but two arrows near each dot will make it too crowded. So we don't draw them normally. Only when we have to think about orientations then we focus on one dot in the picture. What about line integrals? Like our previous discussion on one-dimensional integration. We expect the differential form to be some tick marks or dots on the path we are integrating along. Again, we can have positive and negative dots and use arrows to indicate the orientations of the dots. Then the integral of this differential form is the number of positive dots minus the negative ones. Just like before, if I integrate along a path nearby, I expect a similar series of positive and negative dots on the path. As we vary the path, the dots can be joined together to form curves. In some cases, we can extend this procedure to permeate the whole plane with these curves. In those cases, an integral of the differential form along a path would be the number of positive lines that the path crossed minus the negative ones. If you really want to think about orientations, you can indicate with an arrow on each curve so that when you traverse along a path in a direction against this arrow, you get an extra minus sign. This picture generally works when you zoom in far enough. But for now, let's consider an example dx, which will work throughout the plane. When I integrate dx along a path gamma, I add up the little increments along the x direction. So the integral is how much gamma spans horizontally. Using dots to visualize, we expect them to indicate where we reach one unit of horizontal span. It's a similar story for a slightly different path where the dots should also indicate where one unit of horizontal span is reached. By joining these dots, we obtain a set of vertical lines. Now the integral of dx along a path gamma can instead be computed by counting the number of vertical lines crossed. So this is how dx can be visualized on the plane and the orientation for each line is towards the right meaning that if you integrate towards the right this is counted as positive crossing but integrating to the left is counted as negative crossing. Now that dx looks like this, it shouldn't come as a surprise that dy is represented by a set of horizontal lines and each line is decorated with an upward arrow to indicate its orientation. Now, what about something like 3dx + 2 dy? Let's consider the simplest path being just a horizontal segment. If this segment is 1 unit long and we integrate this 3dx + 2 dy along it, then since this segment doesn't have any y increments, we are left with the dx bit which is three times the horizontal span. In this case with unit length, this is three. So we expect this segment to cross three dots. In a similar fashion for a vertical segment going upwards when we integrate this 3dx + 2dy there aren't any x increments along this vertical path. So we are left with the dy bit which evaluates to two and we expect the vertical segment to cross two dots. With these two simple segments and where the dots are on each of them, we can join them together and even continue throughout the plane filling it with oblique lines.

As for the orientation, it is going to the top right consistent with the positive x direction from dx and the positive y direction from dy. Even though this is a standard way to visualize 3dx + 2d y, we can do something more stupid. This is what 3DX looks like where a segment of unit length in the x direction should cross three lines. And this is what 2dy looks like where a vertical segment of unit length should cross two lines. We can simply combine these two for 3dx + 2dy. The idea is still the same. We integrate the differential form along a path by counting the number of lines it crossed. This picture involves more lines than the other one with the parallel lines. But visualizing the sum of differential forms by combining their respective pictures is useful later on. So there can be multiple ways to represent the same differential form and the visualizations are not unique. Now generally line integrals involve some combinations of dx and dy but the coefficients are functions of x and y. So they depend on where you are at a particular point. It could look like 3dx + 2 dy giving this set of oblique lines. And at some other point it looks like negative dx + 3dy giving another set of oblique lines. Sometimes you can stitch them together but it's not a guarantee. In many situations, we can only picture the differential forms locally at each point. Homework for you. How do we visualize x dy? The answer is in the next video. But a hint is that instead of thinking of 2D y like this where the horizontal lines appear at regular intervals, we can also visualize 2dy with some bunched up double lines. In both pictures, this unit length segment crosses two horizontal lines. This is yet another example of the non-uniqueness of the visualization. So far we have discussed that differential forms in one dimension can be visualized using these tick marks or dots and the integral can be simplified to be counting how many positive dots the interval crosses minus the negative ones. In two dimensions, differential forms can also be visualized by these dots. And the integral can also be computed by counting the number of positive dots covered by the region minus the negative ones. For line integrals in two dimensions, we use positive and negative curves to visualize. And the integral can be computed by counting the number of positive lines crossed minus the negative ones. The generalization to three dimensions is now straightforward. For a threedimensional integral with the differential form f of xyz dx dy dz. The f of xyz still denotes the density of the dots at the point xyz and we can still use positive and negative dots to visualize. This time the orientation is denoted with three arrows which are ordered. Similar to the two-dimensional case, if I swap the order of any two of these arrows, I pick up a minus sign because it's now the opposite orientation. A volume integral can then be computed by counting the number of positive dots minus the negative ones. Just like before, what about a surface integral this time? Like the line integral in 2D, the associated differential form is also a bunch of curves. And to integrate it, we count how many positive lines the surface has intersected

minus the negative ones. If we need to think about orientations, since we are integrating over a surface, the orientation is indicated by two arrows. Lastly, for the line integral, we can visualize the differential form as a bunch of surfaces colored green and red to indicate its sign. A line integral is then computed by counting the net number of surfaces the path has pierced through. The orientation is indicated by an arrow because we are integrating over a one-dimensional path. Even though we have so many different pictures of differential forms depending on the different kinds of integrals, ultimately we want to represent a differential form such that we can compute integrals by counting. Perhaps the net number of dots the object covers or the net number of lines it crosses or even the net number of surfaces it pierces through. This is certainly one way to think about differential forms and I like to call it how it looks. But there is a different equally important way and I call that what it does. This other way of thinking allows us to properly define differential forms. Let's go back to the line integral visualization we had in 2D. At any point P, we can define a function. This function takes in a vector V and spits out a number which we are going to denote as omega at P acting on V. This counts the net number of crossings of a tiny version of the vector epsilon v divided by epsilon as epsilon tends to zero. It sounds a bit complicated. So let's use a simpler example where we have vertical lines separated one unit apart from each other. Suppose the vector v has components v_sub_1 and v_sub_2. Then the tiny version of this vector epsilon v has components epsilon v1 and epsilon v2. Since these lines are vertical, you only cross them when you go horizontally. So the number of crossings is given by the x component of this vector. The limit is pretty straightforward. Simply divide both numerator and denominator by epsilon and we get v1 at the end. Notice that v1 is actually the x component of the original vector v. But remember this picture is supposed to represent the differential form dx. So dx now takes on a new meaning. It is a function that sends any vector to its x component. In this simple case, the function doesn't depend on p the position. But in general, it can. For example, in this illustration, omega at p acting on v should be positive because it crosses the positive lines the correct way. But if P is more to the left then this function returns a negative number because it crosses negative lines. Now taking the epsilon limit means that we only need to care about a small vicinity around the point P because if we zoom in close enough, the picture looks nicer with lines separated at roughly regular intervals. The number of lines this epsilon v vector crosses is roughly epsilon times omega at p acting on v by definition of the limit. If we now add a new vector epsilon w in the picture then by definition the number of crossings is also roughly epsilon * omega at p acting on w. Due to the homogeneity of the lines, even if I translate this vector, it crosses the exact same number of lines. So for the vector sum epsilon * v + w

the net number of crossings should be their sum. But here's the thing. The net number of crossings of this vector is by definition approximately epsilon * omega at p acting on v + w. So this function omega at p is an additive function. It isn't too difficult to convince yourself that when I scale the vector, the net number of crossings also scales accordingly. In symbols, this function omega at P acting on a scaled vector lambda V is lambda times the function acting on the original V. Combining these two facts, we can conclude that omega at p is a linear function. These functions differ depending on where you are. So a differential form omega is a collection of different linear functions at different points. So far we've been talking about differential forms associated with line intervals. What about those associated with surface integrals? The idea is similar. Pick a point P and we define the differential form alpha at this point P to be a function that takes in two input vectors V and W and outputs a limit similar in spirit to the previous case but this time it involves the crossings of the parallelogram formed by the vectors epsilon v and epsilon w. Since we are taking the limit, we only care about the small vicinity around P. And this function is also expected to be nice and bilinear. Bilinear here means linear in both slots. More concretely, if I fix the second slot and the first slot is some linear combinations of two vectors v_sub_1 and v_sub_2, the result is the same as individually applying the function to the two vectors and then take the linear combination. The exact same thing happens when I fix the first slot and take linear combinations of vectors in the second slot. However, there is one new thing in this case which is orientation. In this picture, each line carries orientation indicated by two arrows in order. If the vectors V and W follow the same orientation as that line, it is counted positively. But if I flip V and W around, then they are in opposite orientations as that line. So it is counted negatively. So this function has an extra property of being anti-ymmetric. Swapping the two vectors in the argument picks up a minus sign. So a differential form for a surface integral is a collection of different bilinear anti-ymmetric functions at different points. It's not too surprising that this pattern continues. In general, a differential form is a collection of functions at each point P such that it takes in K vectors and outputs a number. It is linear in each of them. And if you swap any two vectors, you pick up a minus sign. We call the functions that take in one vector one forms. Those that take in two vectors are called two forms. And in general, K forms take in K vectors. Even though we have arrived here using these pictures, which aren't really that rigorous, this definition where a differential form is a collection of functions that are kinear and totally anti-ymmetric is rigorous and is how it's usually defined. In this sense, a differential form is a machine at every point taking in some amount of vectors and spitting out a number. Just note that at different points we need different machines. Using this understanding of differential forms, a line integral can be thought of as first separate this curve into many tiny segments. But now each segment is a

vector all pointing along the direction of integration. Then we apply the machines at different points to the tiny vectors. But remember that these machines are different at different points. After taking in the vectors, the machines spit out some numbers. The line integral is the sum of all these outputs from the different machines. We write this as the integral of the differential form omega along the curve gamma where omega is the collection of machines at different points which is a one form that takes in only one vector at each point. When we do a surface integral, this time we separate the surface into many tiny parallelograms. This time the tiny parallelograms have an orientation in a way that is consistent with the orientation of the integral. In a similar fashion, we take the two vectors into our machines at different points and add them together to compute the integral. Similar to the previous case, we write this as the integral of a differential form alpha over the surface s where alpha is the collection of machines at different points which is a two form that takes in two vectors at each point. So now we have two perspectives on differential forms. how it looks and what it does with the latter giving a more rigorous definition of a differential form. Personally, I prefer this way of thinking mainly because there are some weird cases I care about that can't be visualized neatly using the other method. However, this how it looks perspective is really useful when it comes to wedge products to be discussed later on in the video series. The next video will be about exterior derivatives and the generalized Stokes theorem. I know I've talked about a second channel before, but there will actually be a new video on there pretty soon about a more elegant proof of the Harry Ball theorem in 1946. So, subscribe if you haven't. Also, consider becoming a patron to support these videos and stay tuned for the next one. Bye.