Chapter 1: section 2 - Derivatives and Tangents

Let \(f(x)\) be a smooth map from an open set in \(\mathbb{R}^{n}\) to \(\mathbb{R}^{m}\). \(\forall h \in \mathbb{R}^{n}\), the directional derivative of \(f\) (in direction of h) at \(X\) is defined as \[df_{x}(h) = \lim\limits_{t \to 0} \frac{f(x + th) - f(x)}{t}\] Thus, for each \(x\) in the domain of \(f\), we can define the mapping \(df_{x}(h): \mathbb{R}^{n} \mapsto \mathbb{R}^{m}\) by \(df_{x}(h) \in \mathbb{R}^{m}\). Thus, \(df_{x}\) gives the derivative of \(f\) at \(x\) applied to all \(h \in \mathbb{R}^{n}\).

So if we have \(f(y) = (f_{1}(y), ..., f_{m}(y))\) then \(df_{x}\) may be represented as

thus, \(df_{x}\) is a linear map.

​Chain Rule

- Suppose \(U \subset \mathbb{R}^{n}\), \(V \subset \mathbb{R}^{m}\) are open sets
- and \(f: U \mapsto V\), \(g: V \mapsto \mathbb{R}^{l}\) are smooth maps
Then for each \(x \in U\), \[d(g \circ f)_{x} = dg_{f(x)} \circ df_{x}\] If we have

with derivative maps

"We can use derivatives to identify the linear space that best approximates a manifold X at the point x."

- Suppose \(X \subset \mathbb{R}^{N}\), \(\phi: U \mapsto X\) is a local parametrization around \(x\), and \(U\) is an open set in \(\mathbb{R}^{k}\)
- assume that \(\phi(0) = x\)
- then the best linear approximation to \(\phi: U \mapsto X\) at 0 is \(U \mapsto \phi(0) + d\phi_{0}(u) = x + d\phi_{0}(u)\)

​tangent space (of X at x):

the image of the map \(d\phi_{0}: \mathbb{R}^{k} \mapsto \mathbb{R}^{N}\) denoted \(T_{x}(X)\)

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for a linear transformation \(T: V \mapsto W\)...

image:

the image im\((T)\) is the set \(\{T(\vec{v}): \vec{v} \in V\}\) that is, all vectors in \(W\) which \(= T(\vec{v})\) for some \(\vec{v} \in V\)

kernel:

the kernel ker\((T) = \{\vec{v} \in V : T(\vec{v}) = \vec{0}\}\)

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Thus the tangent space \(T_{x}(X)\) given by the image of \(d\phi_{0}: \mathbb{R}^{k} \mapsto \mathbb{R}^{N}\) is a vector subspace of \(\mathbb{R}^{N}\) and \(x + T_{x}(X)\) is the closest flat approximation to \(X\) through \(x\).

tangent vector:

(to \(X\subset \mathbb{R}^{N}\) at \(x \in X\)): a point \(v\) of \(\mathbb{R}^{N}\) that lies in the vector subspace \(T_{x}(X)\) of \(\mathbb{R}^{N}\)

\(\cdot\) The dimension of the vector space \(T_{x}(X)\) is the dimension \(k\) of \(X\).

"We can now cosntruct the best linear approximation of a smooth map of arbitrary manifolds \(f: X \mapsto Y\) at a point \(x\)."

\(\cdot\) if \(f(x) = y\), the derivative should be a linear transformation of tnagent spaces: \(df_{x}: T_{x}(X) \mapsto T_{y}(Y)\)

Suppose
\(\cdot\) \(\phi:U \mapsto X\) parameterizes \(X\) about \(x\)
\(\cdot\) \(\psi:V \mapsto Y\) parameterizes \(Y\) about \(y\)
\(\cdot\) \(U \subset \mathbb{R}^{k}\) and \(V \subset \mathbb{R}^{l}\)
\(\cdot\) \(\phi(0) = x\) and \(\psi(0) = y\)

for small \(U\), we have:

Taking derivatives and applying the chain rule:

Since \(d\phi_{0}\) is an isomorphism, \(df_{x}\) must be \[df_{x} = d\psi_{0} \circ dh_{0} \circ d\phi_{0}^{-1}\]

​isomorphism: a structure preserving mapping between two structures of same type that can be reversed