A $k$-form is the thing you can integrate over a $k$-dimensional piece of space. That is the whole motivation, and everything technical here — alternating multilinear maps, the wedge product, the $dx_{i_1}\wedge\dots\wedge dx_{i_k}$ basis — exists because signed volume is antisymmetric and forms are built to be exactly as antisymmetric as volume is.
Vector fields, line integrals, surface integrals and flux are four notations for two ideas, and in $\mathbb{R}^3$ the coincidence that a plane has a normal line hides the difference. Forms separate them, and the reward is that grad, curl, div and the three classical integral theorems collapse into one operator and one theorem.
Maxwell's equations are two equations in the language of forms. Symplectic forms carry all of Hamiltonian mechanics; work and flux are $1$- and $2$-forms; thermodynamic relations are statements about which forms are exact.
Grassmann invented the exterior algebra in 1844 and was ignored for decades. Élie Cartan turned it into the calculus of differential forms around 1900, and de Rham's theorem in 1931 showed the forms on a space see its topology — the beginning of a subject the last two units only touch.
A $k$-tensor on $V=\mathbb{R}^n$ is a map $T:V^k\to\mathbb{R}$ linear in each slot with the others fixed. It is alternating if swapping any two arguments flips the sign: $T(\dots,v,\dots,w,\dots)=-T(\dots,w,\dots,v,\dots)$. Write $\Lambda^k(V)$ for the alternating $k$-tensors.
$\dim\Lambda^k(\mathbb{R}^n)=\binom{n}{k}$, with basis the $dx_{i_1}\wedge\dots\wedge dx_{i_k}$ over increasing index tuples $i_1<\dots<i_k$.
An alternating tensor is determined by its values on basis tuples, and antisymmetry means only the increasing tuples carry independent information. For $n=3$ the dimensions are $1,3,3,1$ — and the middle two both being $3$ is the coincidence that lets $\mathbb{R}^3$ get away with vector calculus for so long.
$\det:(\mathbb{R}^n)^n\to\mathbb{R}$ is alternating and $n$-linear in its columns. Since $\dim\Lambda^n(\mathbb{R}^n)=1$, every alternating $n$-tensor is a scalar multiple of it. That one-dimensionality is the real content of "the determinant is the unique normalised signed volume".
For $\omega\in\Lambda^k$ and $\eta\in\Lambda^l$, the wedge $\omega\wedge\eta\in\Lambda^{k+l}$ is the alternation of the tensor product, normalised so that
the determinant of the $k\times k$ matrix of the chosen coordinates of the chosen vectors. Different books insert different constants; only consistency matters.
With $\omega=dx+2dy$ and $\eta=3dy-dz$:
The $dy\wedge dy$ term dies. Expand mechanically, then delete repeats and sort indices, picking up a sign for each transposition.
In $\mathbb{R}^4$, $\omega=dx_1\wedge dx_2+dx_3\wedge dx_4$ gives $\omega\wedge\omega=2\,dx_1\wedge dx_2\wedge dx_3\wedge dx_4\ne0$. This $\omega$ is the standard symplectic form and the non-vanishing of its powers is why phase-space volume is preserved.
A differential $k$-form on an open $U\subseteq\mathbb{R}^n$ is a map assigning to each $x\in U$ an element $\omega(x)\in\Lambda^k$. In coordinates,
and $\omega$ is smooth when its coefficient functions are. A $0$-form is just a function.
For a smooth $g:U\to V$ and a $k$-form $\omega$ on $V$, the pullback $g^{*}\omega$ is the $k$-form on $U$ given by
Push the vectors forward with the derivative, then evaluate. Forms pull back along maps while vector fields do not push forward — the main practical reason forms are the better object.
In practice this means: to pull back, substitute $y_i=g_i(x)$ everywhere and expand $dy_i=\sum_j(\partial g_i/\partial x_j)\,dx_j$ using the wedge rules.
For $g:U\to\mathbb{R}^n$ with $U\subseteq\mathbb{R}^n$,
The change of variables factor was a determinant all along, and here it appears with its sign. Unit 10 needed an absolute value only because unoriented volume had nowhere to record orientation.
With $x=r\cos\theta$, $y=r\sin\theta$: $dx=\cos\theta\,dr-r\sin\theta\,d\theta$ and $dy=\sin\theta\,dr+r\cos\theta\,d\theta$. Wedging,
Three lines of algebra reproduce the polar area element, no geometric picture required.
• Forgetting antisymmetry when sorting indices.
$dy\wedge dx=-dx\wedge dy$, always.
• Writing $\omega\wedge\omega=0$ for even-degree forms. It holds for odd
degree only.
• Mixing normalisation conventions for the wedge between sources. Pick
one and check your text.
• Treating a $2$-form in $\mathbb{R}^3$ as a vector field without saying
which identification you used. It works in $\mathbb{R}^3$ and nowhere
else.
• Trying to push a form forward. Forms pull back; only diffeomorphisms let
you go the other way, via the inverse.
Original problems written for this guide in the style of Spivak and Munkres exercises. Write a full solution before opening the answer.
What is $\dim\Lambda^2(\mathbb{R}^4)$, and list a basis.
Compute $(dx+dy)\wedge(dy+dz)$ in $\mathbb{R}^3$.
Evaluate $dx\wedge dy$ on $v=(1,2,0)$ and $w=(3,1,0)$.
Pull back $x\,dy$ under $g(u,v)=(u^2,uv)$.
Show $v_1\wedge\dots\wedge v_k=0$ exactly when the $v_i$ are dependent.
For a $1$-form $\alpha$ and a $2$-form $\beta$, relate $\alpha\wedge\beta$ and $\beta\wedge\alpha$.
Pull $dx\wedge dy\wedge dz$ back to spherical coordinates.
Why is the identification of $2$-forms with vector fields special to $\mathbb{R}^3$?
Course coverage. This unit is part of the complete MAT257Y1 topic map. It should cover the following items before you move on:
Spivak's chapter on integration on chains develops tensors and forms compactly; Munkres gives a longer treatment of the alternating algebra first. Conventions for the wedge constant differ between texts — match the syllabus.
Antisymmetry gives the dimension count, which makes top forms one-dimensional, which makes the determinant appear under pullback, which is where the change of variables factor comes from. Unit 12 adds the derivative of a form, and Unit 14 integrates it.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Why insist on antisymmetry?
Why is $\Lambda^n(\mathbb{R}^n)$ one-dimensional?
Do forms push forward or pull back?
Why does $\det Jg$ appear when pulling back a top form?