Differential Forms 01: Tangent, Cotangent, and Forms

Differential forms / 01

Tangent, cotangent, and forms

A differential form is not a decorated integral sign. It is a smoothly varying alternating covector, designed to be evaluated on tangent directions.

Calculus root: the differential in one variable

In one-variable calculus the expression

\[df=f'(x)\,dx\]

already separates two roles. The number \(f'(x)\) is the rate of change, while \(dx\) is the infinitesimal input direction. Evaluating \(df\) on a velocity \(v\,\partial_x\) gives

\[df_x(v\,\partial_x)=f'(x)v,\]

which is exactly the directional rate observed along a parametrized curve \(x(t)\) with \(x'(0)=v\). The cotangent vector \(df_x\) is therefore the device that eats tangent vectors and returns first-order change.

Example: ordinary chain rule

If \(f(x)=x^3\) and \(x(t)=2+t^2\), then

\[{d\over dt}f(x(t))=df_{x(t)}(x'(t)\partial_x) =3x(t)^2x'(t).\]

The formula for a 1-form on a manifold is this calculation with several coordinate directions available.

Example: two coordinates branch from the same rule

For \(f(x,y)=x^2y\),

\[df=2xy\,dx+x^2\,dy.\]

On a curve \(r(t)=(x(t),y(t))\),

\[df(r'(t))=2xy\,x'(t)+x^2y'(t),\]

the ordinary derivative of \(f(r(t))\). Tangent and cotangent language is the coordinate-free packaging of this single-variable chain rule.

The PDF begins Part III by defining tangent vectors as derivations and forms as alternating multilinear maps (Part III, Section 4.1, p.154). This is the right starting point: the integral calculus of forms is built from pointwise linear algebra.

Tangent vectors as derivations

Let \(U\subset \mathbb R^n\) be open and \(p\in U\). A tangent vector at \(p\) may be treated as a derivation

\[D:C^\infty(U)\to \mathbb R,\qquad D(fg)=f(p)D(g)+g(p)D(f).\]

The coordinate derivations

\[\partial_i\vert_p(f)={\partial f\over \partial x_i}(p)\]

form a basis of \(T_pU\). Hence every tangent vector is uniquely

\[v=\sum_{i=1}^n v^i\partial_i\vert_p.\]

Example 1: derivation on a polynomial

In \(\mathbb R^2\), let \(p=(1,2)\) and \(v=3\partial_x-\partial_y\). For

\[f(x,y)=x^2y+\sin y,\]

one has

\[v(f)=3(2xy)\vert_{(1,2)}-(x^2+\cos y)\vert_{(1,2)} =12-1-\cos 2.\]

The vector is not a tiny arrow by definition; it is the operator taking directional derivatives at \(p\).

Example 2: tangent vector on a parametrized curve

For \(\gamma(t)=(t,t^2,e^t)\) in \(\mathbb R^3\), the tangent vector at \(t=0\) is

\[\gamma'(0)=\partial_x+0\partial_y+\partial_z.\]

As a derivation it sends \(h(x,y,z)=xz+y^2\) to

\(\gamma'(0)(h)=h_x(0,0,1)+h_z(0,0,1)=1+0=1.\)

Cotangent vectors and coordinate 1-forms

The cotangent space \(T_p^*U\) is the dual vector space of linear functionals on \(T_pU\). The coordinate 1-forms \(dx_i\) are defined by

\[dx_i(\partial_j)=\delta_{ij}.\]

A 1-form on \(U\) is a smooth field of covectors

\[\eta=\sum_{i=1}^n a_i\,dx_i,\qquad a_i\in C^\infty(U).\]

Its value on a vector \(v=\sum v^i\partial_i\) is

\[\eta_p(v)=\sum_{i=1}^n a_i(p)v^i.\]

This row-vector/column-vector picture is useful, but only locally. Under a coordinate change the coefficients and the basis covectors transform together.

Differential of a function

For a smooth function \(f\),

\[df=\sum_i {\partial f\over \partial x_i}\,dx_i.\]

The point of this definition is the identity

\[(df)_p(v)=v(f).\]

Thus \(df\) packages all directional derivatives of \(f\) into a single 1-form.

Definition

A smooth \(k\)-form on \(U\) is a smooth assignment

\[p\longmapsto \omega_p\in \Lambda^k(T_p^*U),\]

where \(\omega_p\) is alternating and multilinear in \(k\) tangent-vector inputs. The space of all smooth \(k\)-forms is denoted \(\Omega^k(U)\).

The low-degree cases matter:

• \(\Omega^0(U)=C^\infty(U)\).
• \(\Omega^1(U)\) consists of covector fields.
• \(\Omega^n(U)\) consists of density-like objects that can be integrated over oriented \(n\)-dimensional regions.
• \(\Omega^k(U)=0\) for \(k>n\).

Local coordinate form

Every \(k\)-form can be written uniquely as

\[\omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\cdots i_k}\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}.\]

This formula should be read as a coordinate expansion, not as a definition of forms. The coordinate-free object is the alternating covector field; the expression above is what one computes with.

Example 3: a 2-form measuring signed area

In \(\mathbb R^2\),

\[\omega=dx\wedge dy\]

acts by

\[\omega(v,w)= \det\begin{pmatrix} dx(v)&dx(w)\\ dy(v)&dy(w) \end{pmatrix}.\]

For \(v=(1,1)\) and \(w=(2,1)\) this gives \(1\cdot 1-2\cdot 1=-1\). Swapping the vectors changes the sign.

Example 4: a top form in three variables

In \(\mathbb R^3\),

\[\omega=(x^2+z)\,dx\wedge dy\wedge dz\]

evaluated at \(p=(1,0,2)\) multiplies oriented volume by \(3\). On vectors \(u,v,w\),

\(\omega_p(u,v,w)=3\det[u\ v\ w].\)

What smoothness means

For a form \(\omega\) to be smooth, its value on smooth vector fields must vary smoothly. In coordinates this is equivalent to the coefficient functions being smooth. For example, the assignment

\[\omega_p= \begin{cases} dx\wedge dy,&x(p)\ge 0,\\ 0,&x(p)<0 \end{cases}\]

is pointwise alternating, but not a smooth 2-form because its coefficient jumps across the line \(x=0\).