A real valued function
f on an
n-dimensional differentiable manifold
M is called
differentiable at a point if it is differentiable in any coordinate chart defined around
p. In more precise terms, if (U,\phi) is a differentiable chart where U is an open set in M containing
p and \phi : U\to {\mathbf R}^n is the map defining the chart, then
f is differentiable at
p if and only if f\circ \phi^{-1} \colon \phi(U)\subset {\mathbf R}^n \to {\mathbf R} is differentiable at \phi(p), that is f\circ \phi^{-1} is a differentiable function from the open set \phi(U), considered as a subset of {\mathbf R}^n, to \mathbf R. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at
p. It follows from the
chain rule applied to the transition functions between one chart and another that if
f is differentiable in any particular chart at
p, then it is differentiable in all charts at
p. Analogous considerations apply to defining
Ck functions, smooth functions, and analytic functions.
Differentiation of functions There are various ways to define the
derivative of a function on a differentiable manifold, the most fundamental of which is the
directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable
affine structure with which to define
vectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors.
Directional differentiation Given a real valued function
f on an
n dimensional differentiable manifold
M, the directional derivative of
f at a point
p in
M is defined as follows. Suppose that γ(
t) is a curve in
M with , which is
differentiable in the sense that its composition with any chart is a
differentiable curve in
Rn. Then the
directional derivative of
f at
p along γ is \left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}. If
γ1 and
γ2 are two curves such that , and in any coordinate chart \phi , \left.\frac{d}{dt}\phi\circ\gamma_1(t)\right|_{t=0}=\left.\frac{d}{dt}\phi\circ\gamma_2(t)\right|_{t=0} then, by the chain rule,
f has the same directional derivative at
p along
γ1 as along
γ2. This means that the directional derivative depends only on the
tangent vector of the curve at
p. Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space.
Tangent vector and the differential A
tangent vector at is an
equivalence class of differentiable curves
γ with , modulo the equivalence relation of first-order
contact between the curves. Therefore, \gamma_1\equiv \gamma_2 \iff \left.\frac{d}{dt}\phi\circ\gamma_1(t)\right|_{t=0} = \left.\frac{d}{dt}\phi\circ\gamma_2(t)\right|_{t=0} in every coordinate chart \phi. Therefore, the equivalence classes are curves through
p with a prescribed
velocity vector at
p. The collection of all tangent vectors at
p forms a
vector space: the
tangent space to
M at
p, denoted ''T'
p'M''. If
X is a tangent vector at
p and
f a differentiable function defined near
p, then differentiating
f along any curve in the equivalence class defining
X gives a well-defined directional derivative along
X: Xf(p) := \left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}. Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative. If the function
f is fixed, then the mapping X\mapsto Xf(p) is a
linear functional on the tangent space. This linear functional is often denoted by
df(
p) and is called the
differential of
f at
p: df(p) \colon T_pM \to {\mathbf R}.
Definition of tangent space and differentiation in local coordinates Let M be a topological n-manifold with a smooth atlas \{(U_\alpha,\phi_\alpha)\}_{\alpha\in A}. Given p\in M let A_p denote \{\alpha\in A:p\in U_\alpha\}. A "tangent vector at p\in M" is a mapping v:A_p\to\mathbb{R}^n, here denoted \alpha\mapsto v_\alpha, such that v_\alpha=D\Big|_{\phi_\beta(p)}(\phi_\alpha\circ\phi_\beta^{-1})(v_\beta) for all \alpha,\beta\in A_p. Let the collection of tangent vectors at p be denoted by T_pM. Given a smooth function f:M\to\mathbb{R}, define df_p:T_pM\to\mathbb{R} by sending a tangent vector v:A_p\to\mathbb{R}^n to the number given by D\Big|_{\phi_\alpha(p)}(f\circ\phi_\alpha^{-1})(v_\alpha), which due to the chain rule and the constraint in the definition of a tangent vector does not depend on the choice of \alpha\in A_p. One can check that T_pM naturally has the structure of a n-dimensional real vector space, and that with this structure, df_p is a linear map. The key observation is that, due to the constraint appearing in the definition of a tangent vector, the value of v_\beta for a single element \beta of A_p automatically determines v_\alpha for all \alpha\in A. The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically : v^i=\widetilde{v}^j\frac{\partial x^i}{\partial\widetilde{x}^j} and df_p(v)=\frac{\partial f}{\partial x^i}v^i. With the idea of the formal definitions understood, this shorthand notation is, for most purposes, much easier to work with.
Partitions of unity One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits
partitions of unity. This distinguishes the differential structure on a manifold from stronger structures (such as analytic and holomorphic structures) that in general fail to have partitions of unity. Suppose that
M is a manifold of class
Ck, where . Let {
Uα} be an open covering of
M. Then a
partition of unity subordinate to the cover {
Uα} is a collection of real-valued
Ck functions
φi on
M satisfying the following conditions: • The
supports of the
φi are
compact and
locally finite; • The support of
φi is completely contained in
Uα for some
α; • The
φi sum to one at each point of
M: \sum_i \phi_i(x) = 1. (Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the
φi.) Every open covering of a
Ck manifold
M has a
Ck partition of unity. This allows for certain constructions from the topology of
Ck functions on
Rn to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of
Rn. Partitions of unity therefore allow for certain other kinds of
function spaces to be considered: for instance
Lp spaces,
Sobolev spaces, and other kinds of spaces that require integration.
Differentiability of mappings between manifolds Suppose
M and
N are two differentiable manifolds with dimensions
m and
n, respectively, and
f is a function from
M to
N. Since differentiable manifolds are topological spaces we know what it means for
f to be continuous. But what does "
f is " mean for ? We know what that means when
f is a function between Euclidean spaces, so if we compose
f with a chart of
M and a chart of
N such that we get a map that goes from Euclidean space to
M to
N to Euclidean space we know what it means for that map to be . We define "
f is " to mean that all such compositions of
f with charts are . Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on
M and
N are selected. However, defining the derivative itself is more subtle. If
M or
N is itself already a Euclidean space, then we don't need a chart to map it to one. == Bundles ==