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Examples of differential forms
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What do differential forms look like?
Here are some examples of differential forms: $$dx, dx \hspace{1pt} dy, 3dx - 2dy.$$ Most of them are already familiar from Calculus.
The set (it is in fact a vector space) of all $k$-forms (in a particular Euclidean space) is denoted by $\Omega^k$.
We will start our exploration with familiar idea of the derivative from calculus 1.
Consider the two notations for the derivative at $a$ of $f$: $$\frac{dy}{dx} = f'(a).$$
As we know, the left hand side is NOT a fraction!
But if it was, then the following result would be immediate: $$dy = f'(a) dx.$$
Whatever this is, we call $f'(a) dx$ a $1$-form.
It is important to realize that here $dx$ is just a certain variable related to $x$. To emphasize this point, the formula can be re-written as $$d_y = f'(a) d_x,$$ or even $$u = f'(a) v,$$ where $f'(a)$ is a linear function (a matrix) and $u$ and $v$ are vectors. This follows from the fact that the derivative is a linear operator as we know from vector calculus.
The issue is related to the best affine approximation. Let's recall that idea.
Example 1: Given a function $f(x)=x^2$, find its best affine approximation at $a=1$.
Since $f'(x)=2x$, we see that $f'(a) = f'(1) = 2$ and, therefore, the best affine approximation of $f$ at $a=1$ is $$T(x)= f(a) + f'(a)(x-a)$$ $$= 1 + 2(x-a).$$ Now we interpret $x-a$ as $dx$.
Then, if we ignore the constant part, we can write $dy = 2dx$. This equation expresses our derivative in terms of these new variables. They are called differentials.
This way we capture the relation between the increment of $x$ and that of $y$ -- close to $a$. Indeed, $y$ grows twice as fast as $x$.
We acquire this information by introducing a new coordinate system $(dy,dx)$. In this coordinate system, the best affine approximation (given by the tangent line) becomes simply a linear function.
One informal way to write forms in a unified fashion is $$\varphi = A_1 dX_1 + A_2 dX_2 + \ldots,$$ where $dX_i$ are "combinations" of $dx$, $dy$, and $dz$.
In any dimension the exterior derivative $df$ of function $f$ is the $1$-form $$df = \nabla f(a) dX.$$ The RHS here should be read as either the fot product or as the product of a matrix and a vector.
Example 2: In ${\bf R}^3$ (and ${\bf R}^2$) we use $x$, $y$, and $z$ for more explicit formulas. Forms are made up of $x$, $y$, $z$, $dx$, $dy$, and $dz$ as follows:
Type of form | Examples in ${\bf R}^2$ | Examples in ${\bf R}^3$ |
---|---|---|
$0$-forms | functions, $x$ and $xy$ | functions, $e^{xy}z$ |
$1$-forms | $\varphi = A dx + B dy$ | $\varphi = A dx + B dy + C dz$ |
$2$-forms | $\varphi = A dx \hspace{1pt} dy$ | $\varphi = A dx \hspace{1pt} dy + B dy \hspace{1pt} dz + C dz \hspace{1pt} dx$ |
$3$-forms | None | $\varphi = A dx \hspace{1pt} dy \hspace{1pt} dz$ |
$4$-forms | None | None |
(Here $A,B,C$ are functions.)
With these all possible (definite) integrals are taken care of: we can compute lengths, surface areas, and volumes!
Note 1: We see there are no $4$-forms in ${\bf R}^2$ or ${\bf R}^3$ because there are no 4-th variable in either space.
Note 2: Keep in mind though that the zero function can be thought of as a k-form for any k.
Generally, not all 1-forms are given by gradients of functions! But we can still see forms as dot products of functions and differentials, for example: $$\phi=(A,B,C) \cdot (dx,dy,dz) = Adx + Bdy + Cdz.$$ Here $(dx,dy,dz)$ isn't really a vector but just a symbolic representation... Following this idea we can write for dimension $n$ $$\phi=V\cdot dX,$$ where $V$ is a vector function of $(x^1,x^2,...,x^n)$ (a vector field) and $dX=(dx^1,dx^2,...,dx^n)$.
Now, this having been very informal, what is a $1$-form mathematically?
Consider, $\varphi = A dx + B dy + C dz$. Here $A,B,C$ are functions of $x,y,z$. Therefore there are a total of 6 variables: $x,y,z,dx,dy,dz$. The rest is just the usual algebra: addition and multiplication. So, we have $$\varphi:{\bf R}^3 \times {\bf R}^3 \rightarrow {\bf R}^3 .$$ We think of $x,y,z$ as locations and $dx,dy,dz$ as directions.
Note that $dx,dy,dz$ appear differently from $x,y,z$ as they are just multiples. In a sense the dependence is linear...
Let's take a closer look at dimension $1$.
Here, all $1$-forms look like this: $$\varphi=A(x) \cdot dx,$$ where $A$ is a continuous (to be able to integrate) function of $x$ multiplied by the second variable called $dx$.
Let's try to plot it. This how we do it:
- first we plot the curve (green) which is the restriction of our function $\varphi$ to a fixed value of $dx$, how about $1$?
- then we observe that $\varphi$ is $0$ if $dx=0$ and plot those points on the $x$ -axis (blue),
- finally we connect these dots to the curve with straight lines (purple).
Looking at the differentials provides another insight. We compare the input of $f$, which is simply the $x$-axis, and that of $df$, which is the $x$-axis and, for each $a$ on the $x$-axis, the corresponding $dx$-axis:
To clarify the terminology: