Differentiable Manifolds/Bases of tangent and cotangent spaces and the differentials

In this section we shall
 * give one base for the tangent and cotangent space for each chart at a point of a manifold,
 * show how to convert representations in one base into another,
 * define the differentials of functions from a manifold to the real line, from an interval to a manifold and from a manifold to another manifold,
 * and prove the chain, product and quotient rules for those differentials.

Some bases of the tangent space
In the following, we will show that these functionals are a basis of the tangent space.

Theorem 2.2: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^n$$ with $$n \ge 1$$ and atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and let $$p \in O$$. For all $$j \in \{1, \ldots, d\}$$:
 * $$\left( \frac{\partial}{\partial \phi_j} \right)_p \in T_p M$$

i. e. the function $$\left( \frac{\partial}{\partial \phi_j} \right)_p : \mathcal C^n(M) \to \mathbb R$$ is contained in the tangent space $$T_p M$$.

Proof:

Let $$\varphi, \vartheta \in \mathcal C^n(M)$$.

1. We show linearity.


 * $$\begin{align}

\left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi + c\vartheta) & = \left( \partial_{x_j} ((\varphi + c\vartheta) \circ \phi^{-1}) \right) (\phi(p)) \\ & = \left( \partial_{x_j} (\varphi \circ \phi^{-1} + c \vartheta \circ \phi^{-1}) \right) (\phi(p)) \\ & = \left( \partial_{x_j} (\varphi \circ \phi^{-1}) + c \partial_{x_j} (\vartheta \circ \phi^{-1})) \right) (\phi(p)) \\ & = \left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi) + c\left( \frac{\partial}{\partial \phi_j} \right)_p (\vartheta) \end{align}$$ From the second to the third line, we used the linearity of the derivative.

2. We show the product rule.


 * $$\begin{align}

\left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi \vartheta) & = \left( \partial_{x_j} ((\varphi \vartheta) \circ \phi^{-1}) \right) (\phi(p)) \\ & = \left( \partial_{x_j} ((\varphi \circ \phi^{-1})(\vartheta \circ \phi^{-1})) \right) (\phi(p)) \\ & = (\varphi \circ \phi^{-1})(\phi(p)) \left( \partial_{x_j} (\vartheta \circ \phi^{-1}) \right) (\phi(p)) + (\vartheta \circ \phi^{-1})(\phi(p)) \left( \partial_{x_j} (\varphi \circ \phi^{-1}) \right) (\phi(p)) \\ & = \varphi(p) \left( \frac{\partial}{\partial \phi_j} \right)_p (\vartheta) + \vartheta(p) \left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi) \end{align}$$ From the second to the third line, we used the product rule of the derivative.

3. It follows from the definition of $$\left( \frac{\partial}{\partial \phi_j} \right)_p$$, that $$\left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi) = 0$$ if $$\varphi$$ is not defined at $$p$$.

Lemma 2.3: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^n$$ with atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, and let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$. If we write $$\phi = (\phi_1, \ldots, \phi_d)$$, then we have for each $$k \in \{1, \ldots, d\}$$, that $$\phi_k \in \mathcal C^n(M)$$.

Proof:

Let $$(U, \theta) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$. Since $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ is an atlas, $$\theta$$ and $$\phi$$ are compatible. From this follows that the function
 * $$\phi|_{U \cap O} \circ \theta|_{O \cap U}^{-1}$$

is of class $$\mathcal C^n$$. But if we denote by $$\pi_k$$ the function
 * $$\pi_k: \mathbb R^d \to \mathbb R, \pi_k(x_1, \ldots, x_d) = x_k$$

, which is also called the projection to the $$k$$-th component, then we have:
 * $$\phi_k|_{U \cap O} \circ \theta|_{O \cap U}^{-1} = \pi_k \circ \phi|_{U \cap O} \circ \theta|_{O \cap U}^{-1}$$

It is not difficult to show that $$\pi_k$$ is contained in $$\mathcal C^\infty(\mathbb R^d, \mathbb R)$$, and therefore the function
 * $$\pi_k \circ \phi|_{U \cap O} \circ \theta|_{O \cap U}^{-1}$$

is contained in $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ as a composition of $$n$$-times continuously differentiable functions (or continuous functions if $$n=0$$).

Lemma 2.4: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^n$$ with $$n \ge 1$$ and atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and let $$p \in O$$. If we write $$\phi = (\phi_1, \ldots, \phi_d)$$ we have:
 * $$\left( \frac{\partial}{\partial \phi_j} \right)_p (\phi_k) = \begin{cases}

1 & j = k \\ 0 & j \neq k \end{cases}$$

Note that due to lemma 2.3, $$\phi_k \in \mathcal C^n(M)$$ for all $$k \in \{1, \ldots, d\}$$, which is why the above expression makes sense.

Proof:

We have:
 * $$\begin{align}

\left( \frac{\partial}{\partial \phi_j} \right)_p (\phi_k) & = \left( \partial_{x_j} (\phi_k \circ \phi^{-1}) \right) (\phi(p)) \\ & = \lim_{y \to 0} \frac{(\phi_k \circ \phi^{-1}) (x_1, \ldots, x_{j-1}, x_j + y, x_{j+1}, \ldots, x_d) - (\phi_k \circ \phi^{-1})(x_1, \ldots, x_d)}{y} \end{align}$$

Further,
 * $$(\phi_k \circ \phi^{-1})(x_1, \ldots, x_d) = x_k$$

and
 * $$(\phi_k \circ \phi^{-1}) (x_1, \ldots, x_{j-1}, x_j + y, x_{j+1}, \ldots, x_d) = \begin{cases}

x_k + y & k = j \\ x_k & k \neq j \end{cases}$$

Inserting this in the above limit gives the lemma.

Theorem 2.5: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^n$$ with $$n \ge 1$$ and atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and let $$p \in O$$. The tangent vectors
 * $$\left( \frac{\partial}{\partial \phi_j} \right)_p \in T_p M, j \in \{1, \ldots, d\}$$

are linearly independent.

Proof:

We write again $$\phi = (\phi_1, \ldots, \phi_d)$$.

Let $$\sum_{j=1}^d a_j \left( \frac{\partial}{\partial \phi_j} \right)_p = 0_p$$. Then we have for all $$k \in \{1, \ldots, d\}$$:
 * $$0 = 0_p(\phi_k) = \sum_{j=1}^d a_j \left( \frac{\partial}{\partial \phi_j} \right)_p (\phi_k) = a_k$$

Lemma 2.6:

Let $$M$$ be a manifold with atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, $$p \in M$$, $$V \subseteq M$$ be open, let $$\mathbf V_p \in T_p M$$ and $$\varphi: V \to \mathbb R, \varphi(q) = c$$ for a $$c \in \mathbb R$$; i. e. $$\varphi$$ is a constant function. Then $$\varphi \in \mathcal C^\infty(M)$$ and $$\mathbf V_p(\varphi) = 0$$.

Proof:

1. We show $$\varphi \in \mathcal C^\infty(M)$$.

By assumption, $$V \subseteq M$$ is open. This means the first part of the definition of a $$\mathcal C^\infty(M)$$ is fulfilled.

Further, for each $$(U, \theta) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and $$x \in \theta(V \cap U)$$, we have:
 * $$\varphi \circ \theta|_{U \cap V}(x) = c$$

This is contained in $$\mathcal C^\infty(\mathbb R^d, \mathbb R)$$.

2. We show that $$\mathbf V_p(\varphi) = 0$$.

We define $$\vartheta: V \to \mathbb R, \vartheta(q) = 1$$. Using the two rules linearity and product rule for tangent vectors, we obtain:
 * $$\mathbf V_p (\varphi) = \mathbf V_p (\vartheta \varphi) = 1 \mathbf V_p(\varphi) + \varphi(p) \mathbf V_p(\vartheta) = \mathbf V_p(\varphi) + \mathbf V_p(\vartheta \varphi(p)) = \mathbf V_p(\varphi) + \mathbf V_p(\varphi)$$

Substracting $$\mathbf V_p(\varphi)$$, we obtain $$\mathbf V_p(\varphi) = 0$$.

Proof:

Let $$U \subseteq M$$ be open, and let $$\varphi: U \to \mathbb R$$ be contained in $$\mathcal C^n(M)$$.

Case 1: $$p \notin U$$.

In this case, $$\mathbf V_p(\varphi) = 0$$ and $$\left( \frac{\partial}{\partial \phi_j} \right)_p(\varphi) = 0$$, since $$\varphi$$ is not defined at $$p$$ and both $$\mathbf V_p$$ and $$\left( \frac{\partial}{\partial \phi_j} \right)_p$$ are tangent vectors. From this follows the formula.

Case 2: $$p \in U$$.

In this case, we obtain that the set $$\phi(U \cap O)$$ is open in $$\mathbb R^d$$ as follows: Since $$\phi: O \to \phi(O)$$ is a homeomorphism by definition of charts, the set $$\phi(U \cap O)$$ is open in $$\phi(O)$$. By definition of the subspace topology, we have $$\phi(U \cap O) = V \cap \phi(O)$$ for a $$V$$ open in $$\mathbb R^d$$. But $$V \cap \phi(O)$$ is open in $$\mathbb R^d$$ as the intersection of two open sets; recall that $$\phi(O)$$ was required to be open in the definition of a chart.

Furthermore, from $$p \in U$$ and $$p \in O$$ it follows that $$p \in U \cap O$$, and therefore $$\phi(p) \in \phi(O \cap U)$$. Since $$\phi(O \cap U)$$ is open, we find an $$\epsilon > 0$$ such that the open ball $$B_\epsilon(\phi(p))$$ is contained in $$\phi(O \cap U)$$. We define $$W := \phi^{-1}(B_\epsilon(\phi(p)))$$. Since $$\phi$$ is bijective, $$W \subseteq U \cap O$$, and since $$\phi$$ is a homeomorphism, in particular continuous, $$W$$ is open in $$O$$ with respect to the subspace topology of $$O$$. From this also follows $$O$$ open in $$M$$, because if $$W$$ is open in $$O$$, then by definition of the subspace topology it is of the form $$V \cap O$$ for an open set $$V \subseteq M$$, and hence it is open as the intersection of two open sets.

We have that $$\varphi|_W: W \to \mathbb R$$, is contained in $$\mathcal C^\infty(M)$$: $$W$$ is an open subset of $$M$$, and if $$(V, \theta) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, then
 * $$\varphi|_{W \cap V} \circ \theta|_{W \cap V}^{-1} = (\varphi|_{U \cap V} \circ \theta|_{U \cap V}^{-1})|_{\theta(W \cap V)}$$,

(check this by direct calculation!), which is contained in $$\mathcal C^\infty(\mathbb R^d, \mathbb R)$$ as the restriction of an arbitrarily often continuously differentiable function.

We now define the function $$F: B_\epsilon(\phi(p)) \to \mathbb R$$, $$F(x) = (\varphi \circ \phi^{-1})(x)$$, and further for each $$x \in B_\epsilon(\phi(p))$$, we define
 * $$\mu_x(\xi) := F(\xi x + (1 - \xi) \phi(p))$$

From the fundamental theorem of calculus, the multi-dimensional chain rule and the linearity of the integral follows for each $$x \in B_\epsilon(\phi(p))$$, that
 * $$\begin{align}

F(x) & = \mu_x(1) \\ & = \mu_x(0) + \int_0^1 \mu_x' (\xi) d\xi \\ & = F(\phi(p)) + \sum_{j=1}^d (x_j - \phi(p)_j) \int_0^1 \partial_{x_j} F(\xi \phi(p) + (1 - \xi) x) d\xi \end{align}$$ If one sets $$x = \phi(q)$$ for $$q \in W$$, one obtains, inserting the definition of $$F$$:
 * $$\varphi(q) = \varphi(p) + \sum_{j=1}^d (\phi(q)_j - \phi(p)_j) \int_0^1 \partial_{x_j} (\varphi \circ \phi^{-1})(\xi \phi(p) + (1 - \xi) \phi(q)) d\xi$$

Now we define the functions
 * $$f_j: W \to \mathbb R, f_j(q) := \int_0^1 \partial_{x_j} (\varphi \circ \phi^{-1})(\xi \phi(p) + (1 - \xi) \phi(q)) d\xi$$

These are contained in $$\mathcal C^\infty(M)$$ since they are defined on $$W$$ which is open, and further, if $$(V, \theta) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, then
 * $$f_j|_{V \cap W} \circ \theta|_{V \cap W}^{-1} = \int_0^1 \partial_{x_j} (\varphi \circ \phi^{-1})(\xi \phi|_{V \cap W} \circ \theta|_{V \cap W}^{-1} + (1 - \xi) \phi|_{V \cap W} \circ \theta|_{V \cap W}^{-1}) d\xi$$

, which is arbitrarily often differentiable by the Leibniz integral rule as the integral of a composition of arbitrarily often differentiable functions on a compact set.

Further, again denoting $$\phi = (\phi_1, \ldots, \phi_d)$$, the functions $$\phi_k$$, $$k \in \{1, \ldots, d\}$$ are contained in $$\mathcal C^\infty(M)$$ due to lemma 2.3.

Since $$\varphi|_W \in \mathcal C^\infty(M)$$, $$\mathbf V_p(\varphi|_W)$$ is defined. We apply the rules (linearity and product rule) for tangent vectors and lemma 2.6 (we are allowed to do so because all the relevant functions are contained in $$\mathcal C^\infty(M)$$), and obtain:
 * $$\begin{align}

\mathbf V_p (\varphi|_W) & = \mathbf V_p \left( \varphi(p) + \sum_{j=1}^d (\phi_j - \phi(p)_j) f_j \right) \\ & = \sum_{j=1}^d \left( \phi_j(p) \mathbf V_p(f_j) + f_j(p) \mathbf V_p(\phi_j) - \phi(p)_j \mathbf V_p(f_j) \right) \\ & = \sum_{j=1}^d f_j(p) \mathbf V_p(\phi_j) \end{align}$$ , since due to our notation it's clear that $$\phi_j(p) = \phi(p)_j$$.

But
 * $$\begin{align}

f_j(p) & = \int_0^1 \partial_{x_j} (\varphi \circ \phi^{-1})(\xi \phi(p) + (1 - \xi) \phi(p)) d\xi \\ & = \int_0^1 \partial_{x_j} (\varphi \circ \phi^{-1})(\phi(p)) d\xi \\ & = \partial_{x_j} (\varphi \circ \phi^{-1})(\phi(p)) \\ & = \left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi) \end{align}$$

Thus we have successfully shown
 * $$\mathbf V_p (\varphi|_W) = \sum_{j=1}^d \mathbf V_p(\phi_j) \left( \frac{\partial}{\partial \phi_j} \right)_p (\varphi)$$

But due to the definition of subtraction on $$\mathcal C^\infty(M)$$, due to lemma 2.6, and due to the fact that the constant zero function is a constant function:
 * $$\mathbf V_p(\varphi|_W - \varphi) = \mathbf V_p(0) = 0$$

Due to linearity of $$\mathbf V_p$$ follows $$0 = \mathbf V_p(\varphi|_W) - \mathbf V_p(\varphi)$$, i. e. $$\mathbf V_p(\varphi|_W) = \mathbf V_p(\varphi)$$. Now, inserting in the above equation gives the theorem.

Together with theorem 2.5, this theorem shows that
 * $$\left\{ \left( \frac{\partial}{\partial \phi_j} \right)_p \Bigg| j \in \{1, \ldots, d\} \right\}$$

is a basis of $$T_p M$$, because a basis is a linearly independent generating set. And since the dimension of a vector space was defined to be the number of elements in a basis, this implies that the dimension of $$T_p M$$ is equal to $$d$$.

Some bases of the cotangent space
Note that $$(d \phi_j)_p$$ is well-defined because of lemma 2.3.

Theorem 2.9: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^\infty$$ and atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and let $$p \in O$$. For all $$j \in \{1, \ldots, d\}$$, $$(d \phi_j)_p$$ is contained in $$T_p M^*$$.

Proof:

By definition, $$d \phi_k$$ maps from $$T_p M$$ to $$\mathbb R$$. Thus, linearity is the only thing left to show. Indeed, for $$\mathbf V_p, \mathbf W_p \in T_p M$$ and $$b \in \mathbb R$$, we have, since addition and scalar multiplication in $$T_p M$$ are defined pointwise:
 * $$\begin{align}

(d \phi_j)_p (\mathbf V_p + b \mathbf W_p) & = (\mathbf V_p + b \mathbf W_p)(\phi_k) \\ & = \mathbf V_p(\phi_k) + b \mathbf W_p(\phi_k) \\ & = (d \phi_j)_p (\mathbf V_p) + b (d \phi_j)_p (\mathbf W_p) \end{align}$$

Lemma 2.10: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^\infty$$ and atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and let $$p \in O$$. For $$j, k \in \{1, \ldots, d\}$$, the following equation holds:
 * $$(d \phi_j)_p \left( \left( \frac{\partial}{\partial \phi_k} \right)_p \right) = \begin{cases}

1 & k = j \\ 0 & k \neq j \end{cases}$$

Proof:

We have:
 * $$(d \phi_j)_p \left( \left( \frac{\partial}{\partial \phi_k} \right)_p \right) = \left( \frac{\partial}{\partial \phi_k} \right)_p (\phi_j) \overset{\text{lemma 2.4}}{=} \begin{cases}

1 & k = j \\ 0 & k \neq j \end{cases}$$

Theorem 2.11: Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^\infty$$ and atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$ and let $$p \in O$$. The cotangent vectors $$(d \phi_j)_p, j \in \{1, \ldots, d\}$$ are linearly independent.

Proof:

Let $$0 = \sum_{j=1}^d a_j (d \phi_j)_p$$, where by $$0$$ we mean the zero of $$T_p M^*$$. Then we have for all $$k \in \{1, \ldots, d\}$$:
 * $$0 = \sum_{j=1}^d a_j (d \phi_j)_p \left( \left( \frac{\partial}{\partial \phi_k} \right)_p \right) \overset{\text{lemma 2.10}}{=} a_k$$

Proof:

Let $$\alpha_p \in T_p M^*$$ and $$\mathbf V_p \in T_p M$$. Due to theorem 2.7, we have
 * $$\mathbf V_p = \sum_{j=1}^d \mathbf V_p(\phi_j) \left( \frac{\partial}{\partial \phi_j} \right)_p$$

Therefore, and due to the linearity of $$\alpha_p$$ (because $$T_p M^*$$ was the space of linear functions to $$\mathbb R$$):
 * $$\begin{align}

\alpha_p(\mathbf V_p) & = \sum_{j=1}^d \mathbf V_p(\phi_j) \alpha_p \left( \left( \frac{\partial}{\partial \phi_j} \right)_p \right) \\ & = \sum_{j=1}^d \alpha_p \left( \left( \frac{\partial}{\partial \phi_j} \right)_p \right) (d \phi_j)_p (\mathbf V_p) \end{align}$$ Since $$\mathbf V_p \in T_p M$$ was arbitrary, the theorem is proven.

From theorems 2.11 and 2.12 follows, as in the last subsection, that
 * $$\left\{ (d \phi_j)_p \big| j \in \{1, \ldots, d\} \right\}$$

is a basis for $$T_p M^*$$, and that the dimension of $$T_p M^*$$ is equal to $$d$$, like the dimension of $$T_p M$$.

Expressing elements of the tangent and cotangent spaces in different bases
If $$M$$ is a manifold, $$p \in M$$ and $$(O, \phi), (U, \theta)$$ are two charts in $$M$$'s atlas such that $$p \in O$$ and $$p \in U$$. Then follows from the last two subsections, that
 * $$\left\{ \left( \frac{\partial}{\partial \phi_j} \right)_p \Bigg| j \in \{1, \ldots, d\} \right\}$$ and $$\left\{ \left( \frac{\partial}{\partial \theta_j} \right)_p \Bigg| j \in \{1, \ldots, d\} \right\}$$ are bases for $$T_p M$$, and
 * $$\left\{ (d \phi_j)_p \big| j \in \{1, \ldots, d\} \right\}$$ and $$\left\{ (d \theta_j)_p \big| j \in \{1, \ldots, d\} \right\}$$ are bases for $$T_p M^*$$.

One could now ask the questions:

If we have an element $$\mathbf V_p$$ in $$T_p M$$ given by $$\mathbf V_p = \sum_{j=1}^d a_j \left( \frac{\partial}{\partial \phi_j} \right)_p$$, then how can we represent $$\mathbf V_p$$ as linear combination of the basis $$\left\{ \left( \frac{\partial}{\partial \theta_j} \right)_p \Bigg| j \in \{1, \ldots, d\} \right\}$$?

Or if we have an element $$\alpha_p$$ in $$T_p M^*$$ given by $$\alpha_p = \sum_{j=1}^d a_j (d \phi_j)_p$$, then how can we represent $$\alpha_p$$ as linear combination of the basis $$\left\{ (d \theta_j)_p \big| j \in \{1, \ldots, d\} \right\}$$?

The following two theorems answer these questions:

Proof:

Due to theorem 2.7, we have for $$j \in \{1, \ldots, d \}$$:
 * $$\left( \frac{\partial}{\partial \phi_j} \right)_p = \sum_{k=1}^d \left( \frac{\partial}{\partial \phi_j} \right)_p(\theta_k) \left( \frac{\partial}{\partial \theta_k} \right)_p$$

From this follows:
 * $$\begin{align}

\mathbf V_p & = \sum_{j=1}^d a_j \left( \frac{\partial}{\partial \phi_j} \right)_p \\ & = \sum_{j=1}^d a_j \sum_{k=1}^d \left( \frac{\partial}{\partial \phi_j} \right)_p(\theta_k) \left( \frac{\partial}{\partial \theta_k} \right)_p \\ & = \sum_{k=1}^d \sum_{j=1}^d a_j \left( \frac{\partial}{\partial \phi_j} \right)_p(\theta_k) \left( \frac{\partial}{\partial \theta_k} \right)_p \end{align}$$

Proof:

Due to theorem 2.12, we have for $$j \in \{1, \ldots, d \}$$:
 * $$(d \phi_j)_p = \sum_{k=1}^d (d \phi_j)_p \left( \left( \frac{\partial}{\partial \phi_k} \right)_p \right) (d \theta_k)_p$$

Thus we obtain:
 * $$\begin{align}

\alpha_p & = \sum_{j=1}^d a_j (d \phi_j)_p \\ & = \sum_{j=1}^d a_j \sum_{k=1}^d (d \phi_j)_p \left( \left( \frac{\partial}{\partial \phi_k} \right)_p \right) (d \theta_k)_p \\ & = \sum_{k=1}^d \sum_{j=1}^d a_j (d \phi_j)_p \left( \left( \frac{\partial}{\partial \phi_k} \right)_p \right) (d \theta_k)_p \end{align}$$

The pullback and the differentials
In this subsection, we will define the pullback and the differential. For the differential, we need three definitions, one for each of the following types of functions:
 * functions from a manifold to another manifold
 * functions from a manifold to $$\mathbb R$$
 * functions from an interval $$I \subseteq \mathbb R$$ to a manifold (i. e. curves)

For the first of these, the differential of functions from a manifold to another manifold, we need to define what the pullback is:

Lemma 2.16: Let $$M$$ be a $$d$$-dimensional and $$N$$ be a $$b$$-dimensional manifold, let $$k \in \mathbb N_0 \cup \{\infty\}$$ and let $$\psi: M \to N$$ be differentiable of class $$\mathcal C^k$$. Then $$\psi$$ is continuous.

Proof:

We show that for an arbitrary $$p \in M$$, $$\psi$$ is continuous on an open neighbourhood of $$p$$. There is a theorem in topology which states that from this follows continuity.

We choose $$(O, \phi)$$ in the atlas of $$M$$ such that $$p \in O$$, and $$(U, \theta)$$ in the atlas of $$N$$ such that $$\psi(p) \in U$$. Due to the differentiability of $$\psi$$, the function
 * $$\theta \circ \psi \circ \phi|_{\phi(O \cap \psi^{-1}(U))}^{-1}$$

is contained in $$\mathcal C^k(\mathbb R^d, \mathbb R^b)$$, and therefore continuous. But $$\phi$$ and $$\theta$$ are charts and therefore homeomorphisms, and thus the function
 * $$\psi|_{O \cap \psi^{-1}(U)}: O \cap \psi^{-1}(U) \to N, \psi = \theta^{-1} \circ \theta \circ \psi \circ \phi|_{O \cap \psi^{-1}(U)}^{-1} \circ \phi|_{O \cap \psi^{-1}(U)}$$

is continuous as the composition of continuous functions.

Lemma 2.17: Let $$M, N$$ be two manifolds, let $$\psi: M \to N$$ be differentiable of class $$\mathcal C^k$$, and let $$\varphi \in \mathcal C^k(N)$$ be defined on the open set $$U \subseteq N$$. In this case, the function $$\varphi \circ |_{\psi^{-1}(U)}$$ is contained in $$\mathcal C^k(M)$$; i. e. the pullback with respect to $$\psi$$ really maps to $$\mathcal C^k(M)$$.

Proof:

Since $$\psi$$ is continuous due to lemma 2.16, $$\psi^{-1}(U)$$ is open in $$M$$. Thus $$\varphi \circ \psi|_{\psi^{-1}(U)}$$ is defined on an open set.

Let $$(O, \phi)$$ be an arbitrary element of the atlas of $$M$$ and let $$x \in \phi(O)$$ be arbitrary. We choose $$(V, \theta)$$ in the atlas of $$N$$ such that $$\psi(\phi^{-1}(x)) \in V$$. The function
 * $$(\varphi \circ \psi|_{\psi^{-1}(U)} \circ \phi|_{\psi^{-1}(U) \cap O}^{-1})|_{\phi(\psi^{-1}(U \cap V) \cap O)} = \varphi|_{\psi(\psi^{-1}(U \cap V) \cap O)} \circ \theta|_{\psi(\psi^{-1}(U \cap V) \cap O)}^{-1} \circ \theta|_{\psi(\psi^{-1}(U \cap V) \cap O)} \circ \psi|_{\psi^{-1}(U \cap V) \cap O} \circ \phi|_{\phi(\psi^{-1}(U \cap V) \cap O)}^{-1}$$

is $$k$$-times continuously differentiable (or continuous if $$k=0$$) at $$x$$ as the composition of two $$k$$ times continuously differentiable (or continuous if $$k=0$$) functions. Thus, the function
 * $$\varphi \circ \psi|_{\psi^{-1}(U)} \circ \phi|_{\psi^{-1}(U) \cap O}^{-1}$$

is $$k$$-times continuously differentiable (or continuous if $$k=0$$) at every point, and therefore contained in $$\mathcal C^k(\mathbb R^d, \mathbb R)$$.

Theorem 2.19:

Let $$M, N$$ be two manifolds of class $$\mathcal C^n$$, let $$\psi: M \to N$$ be differentiable of class $$\mathcal C^n$$ and let $$p \in M$$. We have $$\mathbf V_p \circ \psi^* \in T_p N$$; i. e. the differential of $$\psi$$ at $$p$$ really maps to $$T_p N$$.

Proof:

Let $$O, U \subseteq M$$ be open, $$\varphi: O \to \mathbb R, \vartheta: U \to \mathbb R \in \mathcal C^n(M)$$ and $$c \in \mathbb R$$ be arbitrary. In the proof of the following, we will use that for all open subsets $$V \subseteq O$$, $$\mathbf V_p(\varphi|_V) = \mathbf V_p(\varphi)$$ (which follows from the linearity of $$\mathbf V_p$$).

1. We prove linearity.


 * $$\begin{align}

(\mathbf V_p \circ \psi^*)(\varphi + c \vartheta) & = \mathbf V_p (\psi^*(\varphi + c \vartheta)) \\ & = \mathbf V_p((\varphi + c \vartheta) \circ \psi|_{\psi^{-1}(O \cap U)}) \\ & = \mathbf V_p(\varphi|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)} + c \vartheta|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)}) \\ & = \mathbf V_p(\varphi|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)}) + c \mathbf V_p(\vartheta|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)}) \\ & = \mathbf V_p(\psi^*(\varphi)) + c \mathbf V_p(\psi^*(\vartheta)) \\ & = (\mathbf V_p \circ \psi^*)(\varphi) + c (\mathbf V_p \circ \psi^*)(\vartheta) \end{align}$$

2. We prove the product rule.


 * $$\begin{align}

(\mathbf V_p \circ \psi^*)(\varphi \vartheta) & = \mathbf V_p (\psi^*(\varphi \vartheta)) \\ & = \mathbf V_p ((\varphi|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)})(\vartheta|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)})) \\ & = (\varphi|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)})(p) \mathbf V_p(\vartheta|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)}) + (\vartheta|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)})(p) \mathbf V_p(\varphi|_{O \cap U} \circ \psi|_{\psi^{-1}(O \cap U)}) \\ & = \varphi(\psi(p)) \mathbf V_p(\psi^* \vartheta) + \vartheta(\psi(p)) \mathbf V_p(\psi^* \varphi) \\ & = \varphi(\psi(p)) (\mathbf V_p \circ \psi^*)(\vartheta) + \vartheta(\psi(p)) (\mathbf V_p \circ \psi^*)(\varphi) \end{align}$$

Theorem 2.22: Let $$M$$ be a manifold of class $$\mathcal C^n$$, $$n \ge 1$$, let $$I \subseteq \mathbb R$$ be an interval, let $$y \in I$$ and let $$\gamma: I \to M$$ be a differentiable curve of class $$\mathcal C^n$$. Then $$\varphi \circ \gamma$$ is contained in $$\mathcal C^n(\mathbb R, \mathbb R)$$ for every $$\varphi \in \mathcal C^n(M)$$ and $$\gamma'_y$$ is a tangent vector of $$M$$ at $$\gamma(y)$$.

Proof:

1. We show $$\forall \varphi \in \mathcal C^n(M) : \circ \gamma \in \mathcal C^n(\mathbb R, \mathbb R)$$

Let $$x \in I$$ be arbitrary, and let $$U$$ be the set where $$\varphi$$ is defined ($$U$$ is open by the definition of $$\mathcal C^n(M)$$ functions. We choose $$(O, \phi)$$ in the atlas of $$M$$ such that $$\gamma(x) \in O$$. Then the function
 * $$(\varphi \circ \gamma)|_{\gamma^{-1}(O \cap U) \cap I} = \varphi \circ \phi^{-1} \circ \phi \circ \gamma|_{\gamma^{-1}(O \cap U) \cap I}$$

is contained in $$\mathcal C^n(\mathbb R, \mathbb R)$$ as the composition of two $$n$$ times continuously differentiable (or continuous if $$n=0$$) functions.

Thus, $$\varphi \circ \gamma$$ is $$n$$ times continuously differentiable (or continuous if $$n=0$$) at every point, and hence $$n$$ times continuously differentiable (or continuous if $$n=0$$).

2. We show that $$\gamma'_y \in T_{\gamma(y)} M$$ in three steps:

Let $$\varphi, \vartheta \in \mathcal C^n(M)$$ and $$c \in \mathbb R$$.

2.1 We show linearity.

We have:
 * $$\begin{align}

\gamma'_y (\varphi + c\vartheta) & = ((\varphi + c\vartheta) \circ \gamma)' (y) \\ & = (\varphi \circ \gamma + c \vartheta \circ \gamma)' (y) \\ & = (\varphi \circ \gamma)'(y) + c (\vartheta \circ \gamma)' (y) \\ & = \gamma'_y (\varphi) + c \gamma'_y (\vartheta) \end{align}$$

2.2 We prove the product rule.


 * $$\begin{align}

\gamma'_y (\varphi \vartheta) & = ((\varphi \vartheta) \circ \gamma)' (y) \\ & = ((\varphi \circ \gamma)(\vartheta \circ \gamma))' (y) \\ & = (\varphi \circ \gamma)(y) (\vartheta \circ \gamma)' (y) + (\vartheta \circ \gamma)(y) (\varphi \circ \gamma)' (y) \\ & = \varphi(\gamma(y)) \gamma'_y (\vartheta) + \vartheta(\gamma(y)) \gamma'_y (\varphi) \end{align}$$

2.3 It follows from the definition of $$\gamma'_y$$ that $$\gamma'_y(\varphi)$$ is equal to zero if $$\varphi$$ is not defined at $$\gamma(y)$$.

Linearity of the differential for Ck(M), product, quotient and chain rules
In this subsection, we will first prove linearity and product rule for functions from a manifold to $$\mathbb R$$.

Proof:

1. We show that $$\varphi + c \vartheta \in \mathcal C^k(M)$$.

Let $$U$$ be the (open as intersection of two open sets) set on which $$\varphi + c \vartheta$$ is defined, and let $$(O, \phi)$$ be contained in the atlas of $$M$$. The function
 * $$(\varphi + c \vartheta)|_{O \cap U} \circ \phi|_{O \cap U}^{-1} = \varphi|_{O \cap U} \circ \phi|_{O \cap U}^{-1} + c \vartheta|_{O \cap U} \circ \phi|_{O \cap U}^{-1}$$

is contained in $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ as the linear combination of two $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ functions.

2. We show that $$d (\varphi + c \vartheta) = d \varphi + c d \vartheta$$.

For all $$p \in M$$ and $$\mathbf V_p \in T_p M$$, we have:
 * $$d (\varphi + c \vartheta)_p (\mathbf V_p) = \mathbf V_p(\varphi + c \vartheta) = \mathbf V_p(\varphi) + c \mathbf V_p(\vartheta) = d \varphi_p (\mathbf V_p) + c d \vartheta_p (\mathbf V_p)$$

Remark 2.24: This also shows that for all $$\varphi \in \mathcal C^n(M)$$, $$d \varphi_p \in T_p M^*$$.

Proof:

1. We show that $$\varphi \vartheta \in \mathcal C^k(M)$$.

Let $$U$$ be the (open as intersection of two open sets) set on which $$\varphi \vartheta$$ is defined, and let $$(O, \phi)$$ be contained in the atlas of $$M$$. The function
 * $$(\varphi \vartheta)|_{O \cap U} \circ \phi|_{O \cap U}^{-1} = \varphi|_{O \cap U} \circ \phi|_{O \cap U}^{-1} \vartheta|_{O \cap U} \circ \phi|_{O \cap U}^{-1}$$

is contained in $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ as the product of two $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ functions.

2. We show that $$d (\varphi \vartheta) = \varphi d \vartheta + \vartheta d \varphi$$.

For all $$p \in M$$ and $$\mathbf V_p \in T_p M$$, we have:
 * $$d (\varphi \vartheta)_p (\mathbf V_p) = \mathbf V_p (\varphi \vartheta) = \varphi(p) \mathbf V_p (\vartheta) + \vartheta(p) \mathbf V_p (\varphi) = \varphi(p) d \vartheta_p + \vartheta(p) d \varphi_p$$

Proof:

1. We show that $$\frac{\varphi}{\vartheta} \in \mathcal C^n(M)$$:

Let $$U$$ be the (open as the intersection of two open set) set on which $$\frac{\varphi}{\vartheta}$$ is defined, and let $$(O, \phi)$$ be in the atlas of $$M$$ such that $$O \cap U \neq \emptyset$$. The function
 * $$\frac{\varphi}{\vartheta} \big|_{O \cap U} \circ \phi|_{O \cap U}^{-1} = \frac{\varphi|_{O \cap U} \circ \phi|_{O \cap U}^{-1}}{\vartheta|_{O \cap U} \circ \phi|_{O \cap U}^{-1}}$$

is contained in $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ as the quotient of two $$\mathcal C^n(\mathbb R^d, \mathbb R)$$ from which the function in the denominator vanishes nowhere.

2. We show that $$d \left( \frac{\varphi}{\vartheta} \right) = \frac{\vartheta d \varphi - \varphi d \vartheta}{\vartheta^2}$$:

Choosing $$\varphi$$ as the constant one function, we obtain from 1. that the function $$\frac{1}{\vartheta}$$ is in $$\mathcal C^n(M)$$. Hence follows from the product rule:
 * $$0 = d \left( \vartheta \frac{1}{\vartheta} \right) = \vartheta d \left( \frac{1}{\vartheta} \right) + \frac{1}{\vartheta} d \vartheta$$

which, through equivalent transformations, can be transformed to
 * $$d \left( \frac{1}{\vartheta} \right) = -\frac{d \vartheta}{\vartheta^2}$$

From this and from the product rule we obtain:
 * $$d \left( \varphi \frac{1}{\vartheta} \right) = \frac{1}{\vartheta} d \varphi -\frac{\varphi d \vartheta}{\vartheta^2} = \frac{\vartheta d \varphi - \varphi d \vartheta}{\vartheta^2}$$

Proof:

1. We already know that $$\varphi \circ \psi = \psi^* \varphi$$ is differentiable of class $$\mathcal C^n$$; this is what lemma 2.17 says.

2. We prove that $$d(\psi^* \varphi)_p = d \varphi_{\psi(p)} \circ d \psi_p$$.

Let $$\mathbf V_p \in T_p M$$. Then we have:
 * $$\begin{align}

(d \varphi_{\psi(p)} \circ d \psi_p)(\mathbf V_p) & = d \varphi_{\psi(p)} (d \psi_p (\mathbf V_p)) \\ & = d \varphi_{\psi(p)} (\mathbf V_p \circ \psi) \\ & = (\mathbf V_p \circ \psi^*)(\varphi) \\ & = \mathbf V_p (\psi^*(\varphi)) \\ & = \mathbf V_p (\varphi \circ \psi) \\ & = d(\varphi \circ \psi)_p (\mathbf V_p) \end{align}$$

Now, let's go on to proving the chain rule for functions from manifolds to manifolds. But to do so, we first need another theorem about the pullback.

Theorem 2.28: Let $$M, N, K$$ be three manifolds, and let $$\psi: M \to N$$ and $$\chi: N \to K$$ be two functions differentiable of class $$\mathcal C^k$$. Then
 * $$(\chi \circ \psi)^* = \psi^* \circ \chi^*$$

Proof: Let $$\varphi \in \mathcal C^k(K)$$. Then we have:
 * $$\begin{align}

(\chi \circ \psi)^* (\varphi) & = \varphi \circ (\chi \circ \psi) \\ & = (\varphi \circ \chi) \circ \psi \\ & = \chi^* (\varphi) \circ \psi \\ & = \psi^* (\chi^*(\varphi)) \end{align}$$

Proof:

1. We prove that $$\chi \circ \psi$$ is differentiable of class $$\mathcal C^k$$.

Let $$(O, \phi)$$ be contained in the atlas of $$M$$ and let $$(U, \theta)$$ be contained in the atlas of $$K$$ such that $$O \cap \psi^{-1}(\chi^{-1}(U)) \neq \emptyset$$, and let $$x \in \phi^{-1}(O) \cap \psi^{-1}(\chi^{-1}(U))$$ be arbitrary. We choose $$(V, \eta)$$ in the atlas of $$N$$ such that $$\psi(\phi(x)) \in V$$.

We have $$\psi(\phi(x)) \in V \cap \chi^{-1}(U)$$; indeed, $$\psi(\phi(x)) \in V$$ due to the choice of $$(V, \phi)$$ and $$\psi(\phi(x)) \in \chi^{-1}(U)$$ because $$\phi(x) \in \psi^{-1}(\chi^{-1}(U))$$. Further, we choose $$W := O \cap \psi^{-1}(V \cap \chi^{-1}(U))$$. Then the function
 * $$\theta^{-1} \circ (\chi \circ \psi) \circ \phi|_W^{-1} = \theta^{-1} \circ \chi \circ \eta^{-1} \circ \eta \circ \psi|_W \circ \phi|_W^{-1}$$

is contained in $$\mathcal C^n(\mathbb R^d, \mathbb R^d)$$ as the composition of two $$\mathcal C^n(\mathbb R^d, \mathbb R^d)$$ functions.

Thus, $$\theta^{-1} \circ (\chi \circ \psi) \circ \phi|_{O \cap \psi^{-1}(\chi^{-1}(U))}^{-1}$$ is $$n$$ times continuously differentiable (or continuous if $$n=0$$) at every point, and thus $$n$$ times continuously differentiable (or continuous if $$n=0$$).

2. We prove that $$\forall p \in M : d(\chi \circ \psi)_p = d \chi_{\psi(p)} \circ d \psi_p$$.

For all $$p \in M$$ and $$\mathbf V_p \in T_p M$$, we have:
 * $$\begin{align}

(d \chi_{\psi(p)} \circ d \psi_p)(\mathbf V_p) & = d \chi_{\psi(p)} (d \psi_p(\mathbf V_p)) \\ & = d \chi_{\psi(p)} (\mathbf V_p \circ \psi^*) \\ & = \mathbf V_p \circ \psi^* \circ \chi^* \\ & \overset{\text{theorem 2.26}}{=} \mathbf V_p \circ (\chi \circ \psi)^* \\ & = d(\chi \circ \psi)_p (\mathbf V_p) \end{align}$$

Proof:

1. Among another thing, theorem 2.22 states that $$\varphi \circ \gamma$$ is contained in $$\mathcal C^n(\mathbb R, \mathbb R)$$.

2. We show that $$(\varphi \circ \gamma)'(y) = d \varphi_{\gamma(y)}(\gamma'_y)$$:


 * $$\begin{align}

d \varphi_{\gamma(y)} (\gamma'_y) & = \gamma'_y(\varphi) \\ & = (\varphi \circ \gamma)'(y) \end{align}$$

Intuition behind the tangent space
In this section, we want to prove that what we defined as the tangent space is isomorphic to a space whose elements are in analogy to tangent vectors to, say, tangent vectors of a function $$f: \mathbb R \to \mathbb R$$.

We start by proving the following lemma from linear algebra:

Proof:

We only prove that $$T$$ is a vector space isomorphism; that $$S$$ and $$L$$ are also vector space isomorphisms will follow in exactly the same way.

From $$L \circ S \circ T = \text{Id}_{\mathbf V}$$ and $$T \circ L \circ S = \text{Id}_{\mathbf W}$$ follows that $$L \circ S$$ is the inverse function of $$T$$.