If we consider a general mapping between manifolds Φ:Mm→Nn, we can choose charts αM:M→Rm and αN:N→Rn, with coordinate functions xμ and yν, so that the mapping αN∘Φ:M→Rn can be represented by n functions Φν:M→R. This allows us to write down an expression for the induced tangent mapping or differential (aka pushforward, derivative) dΦ:TM→TN (also denoted TΦ or Φ∗ or sometimes simply Φ if it is clear the argument is a tangent vector). For a tangent vector v=vμ∂/∂xμ at a point p∈M we define
dΦ(v)|p≡vμ∂Φλ∂xμ∂∂yλ|Φ(p).
This definition can be shown to be coordinate-independent and to follow our intuitive expectation that mapped tangent vectors stay tangent to mapped curves. If M=N and Φ is the identity, dΦ is just the vector component transformation from the previous section on tangent vectors. The matrix JΦ(x)≡∂Φν/∂xμ is called the Jacobian matrix (AKA Jacobian). For the parametrized curve C:R→Nn, we define the tangent to the curve at t∈R to be
˙C(t)≡dC(∂∂x)|t=∂Cλ∂x∂∂yλ|C(t),
which is also denoted dC(t)/dt and coincides with the Euclidean tangent to a curve if N=Rn.
If Φ is a diffeomorphism, dΦ is an isomorphism between the tangent spaces at every point in M. The inverse function theorem says that the converse is true locally: if dΦp is an isomorphism at p∈M, then Φ is locally a diffeomorphism. In particular, this means that if in some coordinates the Jacobian is nonsingular, then αN∘Φ∘α−1M represents a locally valid coordinate transformation and Φν=yν.
A mapping between manifolds Φ:Mm→Nn also can be used to naturally define the pullback of a form Φ∗:ΛkN→ΛkM by Φ∗φ(v1,…,vk)=φ(dΦ(v1),…,dΦ(vk)), where the name indicates that a form on N can be “pulled back” to M using Φ. Note that the composition of pullbacks is then Ψ∗Φ∗φ=(ΦΨ)∗φ.
Note that for a mapping f:M→R, we have df:TM→TR≅R, so that df(v)=vμ∂f/∂xμ=v(f), the directional derivative of f. Let us apply this to the coordinate function x1:M→R. Then we have dx1(v)=vμ∂x1/∂xμ=v1, so that in particular dxv(∂/∂xμ)=δvμ, i.e. dxμ is in fact the dual frame to ∂/∂xμ. Thus in a given coordinate system, we can write a general tensor of type (m,n) as
T=Tμ1…μmν1…νn∂∂xμ1⊗⋯⊗∂∂xμm⊗dxν1⊗⋯⊗dxνn.
In particular, the metric tensor is often written
ds2≡g=gμνdxμdxν,
where the Einstein summation convention is used and the tensor symbol omitted. A general k-form φ∈ΛkM can then be written as
φ=∑μ1<⋯<μkφμ1…μkdxμ1∧⋯∧dxμk.
From either the tangent mapping definition or the behavior of the exterior product under a change of basis, we see that under a change of coordinates we have
dyμ1∧⋯∧dyμk=det(∂yν∂xμ)dxμ1∧⋯∧dxμk.
This is the familiar Jacobian determinant (like the Jacobian matrix, also often called the Jacobian) that appears in the change of coordinates rule for integrals from calculus, and explains the name of the volume form as defined previously in terms of the exterior product.
In summary, the differential d has a single definition, but is used in several different settings that are not related in an immediately obvious way.
Construct | Argument | Other names | Other symbols |
---|---|---|---|
dΦ:TM→TN | Φ:M→N | Tangent mapping | TΦ, Φ∗, Φ |
df:TM→R | f:M→R | Directional derivative | v(f), dvf, ∇vf |
dxμ:TM→R | xμ:M→R | Dual frame to ∂/∂xμ | βμ |