Differentiabilty of vector valued functions of R

Differentiability

Def: Let $f : D \subseteq R \to R^{n}$ is differentiable on $x_{0} \in D$ if the limit exists:

lim_{x \to a} \frac{f (x) - f (x_{0})}{x - x_{0}}

then its value is the derivative of $f$ on $x_{0}$ , usually denoted as $f^{'} (x_{0})$ .

Th: Let $f : D \subseteq R \to R^{n}$ is differentiable on $x_{0} \in D$ , iff for each $1 \leq i \leq n$ component functions $f_{i}$ is differentiable at $x_{0}$ . If $f$ is differentiable at $x_{0}$ , then:

\begin{aligned} f^{'} (t) & = (\begin{array}{c} f_{1}^{'} (t) \\ f_{2}^{'} (t) \\ ⋮ \\ f_{n}^{'} (t) \end{array}) \end{aligned}

Def: Let $f : D \subseteq R \to R^{n}$ is differentiable on $x_{0} \in D$

The derivative $f^{'} (x_{0})$ will also be called the tangent vector of $f$ at $x_{0}$
If $f^{'} (x_{0}) \neq 0$ , then the straight line in $R^{n}$ through $f (x_{0})$ with direction $f^{'} (x_{0})$ will be called the ****tangent line of $f$ at $x_{0} .$ It is the set ${f (x_{0}) + t f^{'} (x_{0}) ∣ t \in R}$

Def: Let $f : D \subseteq R \to R^{n}$ is differentiable if $f$ is differentiable at each $x \in D$ . Then the function $f^{'} : D \subseteq R \to R^{n}$ whose image is $f^{'} (x)$ for every $x \in D$ .

Th: Let $f, g : D \subseteq R \to R^{n}$ and $ϕ : D \subseteq R \to R$ be differentiable at $x_{0} \in D .$ Then:

$f + g$ is differentiable at $x_{0}$ , and $(f + g)^{'} (x_{0}) = f^{'} (x_{0}) + g^{'} (x_{0})$
$f \cdot g$ is differentiable at $x_{0}$ , and $(f \cdot g)^{'} (x_{0}) = f^{'} (x_{0}) \cdot g (x_{0}) + f (x_{0}) \cdot g^{'} (x_{0})$ Alternatively it can be expressed using the inner product notation: ${⟨ f, g ⟩}^{'} (x_{0}) = ⟨ f^{'} (x_{0}), g (x_{0}) ⟩ + ⟨ f (x_{0}), g^{'} (x_{0}) ⟩$
$ϕ f$ is differentiable at $x_{0}$ , and $(ϕ f)^{'} (x_{0}) = ϕ (x_{0}) f^{'} (x_{0}) + ϕ^{'} (x_{0}) f (x_{0})$
If $n = 3$ , then $f \times g$ is differentiable at $x_{0}$ , and $(f \cdot g)^{'} (x_{0}) = f^{'} (x_{0}) \cdot g (x_{0}) + f (x_{0}) \cdot g^{'} (x_{0})$

Th: Let $f : D \subseteq R \to R^{n}$ is differentiable, and $∥ f (x) ∥$ is constant for every $x \in D$ , then $f (x)$ and $f^{'} (x)$ are orthogonal for every $x \in D$ .

Def: Let $f : D \subseteq R \to R^{n}$ is differentiable, and $f^{'}$ is also continuous, then $f$ is a $C^{1}$ function on $D$ , or $f \in C^{1} (D)$ . If the $k$ th derivative is continuous then $f$ is a $C^{k}$ function on $D$ , or $f \in C^{k} (D)$ .

Def: A subset $C \subseteq R^{n}$ is a curve if there is a $C^{1}$ function $f : D \subseteq R \to R^{n}$ , where $D$ is an interval where $f (D) = C$ . The function $f$ is called a $C^{1}$ parametrization of $C$ .

Def: Let $f : D \subseteq R \to R^{n}$ and $h : E \subseteq R \to R^{n}$ be $C^{1}$ parametrization of the curve $C$ . Then $h$ is equivalent to $f$ if there is a differentiable function $ϕ$ from $E$ onto $D$ such that:

$h = f \circ ϕ$
$ϕ^{'} (x) > 0$ for all $x \in E$ or $ϕ^{'} (x) < 0$ for all $x \in E$

In the case that $ϕ$ is strictly increasing, then we say that $f$ and $h$ are properly equivalent. In the other hand, if $ϕ$ is strictly decreasing, the we say that $f$ and $h$ are equivalent with opposite orientations.

Th: In this sense this type of equivalence does form an equivalence relation on the class of all $C^{1}$ parametrization of $C$ .

Chain Rule: Let $ϕ : E \subseteq R \to R$ and $f : D \subseteq R \to R^{n}$ be differentiable functions and let $ϕ (E) \subseteq D$ . Then $f \circ ϕ$ is differentiable and for each $x \in E$ :

(f \circ ϕ) (x) = f^{'} (ϕ (x)) ϕ^{'} (x)

Th: Let $f : D \subseteq R \to R^{n}$ and $h : E \subseteq R \to R^{n}$ be equivalent $C^{1}$ parametrizations of a curve $C$ in $R^{n}$ related by a differentiable function $ϕ : E \to D$ . Let $h = f \circ ϕ$ and $ϕ$ be strictly monotonic, then whenever $ϕ (u) = t$ then:

The tangent vector of $f$ at $t$ , if non-zero, is parallel to the tangent vector of $h$ at $u$ . The sign depends if $f$ and $h$ have the same orientation or opposite.

Def: A function $f : D \subseteq R \to R^{n}$ is said to be smooth/regular if it is $C^{1}$ and if $f^{'} (t) \neq 0$ for all $t \in D$

Def: A curve $C$ in $R^{n}$ is a (smooth) simple arc if $C$ has a $1 - 1$ function (smooth) $C^{1}$ parametrization of the form $f : [a, b] \subseteq R \to R^{n}$ .

The points $f (a)$ and $f (b)$ are called the endpoints of the arc. The function $f$ is called a simple parametrization of $C$ .

Th: Let $C$ be simple arc in $R^{n}$ simply parametrized by a smooth injective function $f : [a, b] \to R^{n}$ . Then, any smooth parametrization $h : [c, d] \to R^{n}$ of $C$ is injective and equivalent to $f$ .

Cor: Let $q$ be a point in a smooth simple arc $C \subseteq R^{n}$ . Then all smooth parametrizations of $C$ associate the same tangent line with $q$ .

Def: Let $C$ be smooth simple arc in $R^{n}$ parametrized by smooth function $f : [a, b] \to R^{n}$ . The pair $(C, [f])$ , where $[f]$ is the set of all smooth parametrizations of $C$ which are properly equivalent to $f$ , is called an oriented smooth simple arc.

Th: Let $f : [a, b] \to R^{n}$ and $h : [c, d] \to R^{n}$ be $1 - 1$ parametrizations of a simple arc $C$ in $R^{n}$ . Then there is unique $ϕ$ from $[a, b]$ onto $[c, d]$ such that $h = f \circ ϕ$ . Moreover $ϕ$ is continuous and strictly monotonic. In particular:

If $ϕ$ is strictly increasing then $f (a) = h (c)$ and $f (b) = h (d)$
If $ϕ$ is strictly decreasing then $f (a) = h (d)$ and $f (b) = h (c)$

Def: A function $f : [a, b] \to R^{n}$ is said to be piecewise $C^{1}$ (piecewise smooth) if there is a partition $P = {x_{i}}_{i = 0}^{m}$ of $[a, b]$ such that for each $1 \leq k \leq m$ the function $f$ is restricted to the subinterval $[x_{k - 1}, x_{k}]$ is $C^{1}$ (smooth).

Note that in the cases of $x_{k}$ the derivatives might not exists, but their left and right derivative must exist.

Differentiability and linear approximation

Def: Let the set $f : D \subseteq R \to R^{n}$ , and $p \in D$ . Let’s consider the function $δ f_{p} : D_{p} \subseteq R \to R^{n}$ where $D_{p} = {h \in R ∣ p + h \in D}$ ,

δ f_{p} (h) := f (p + h) - f (p)

Th: A function $f : D \subseteq R \to R^{n}$ is differentiable at $p \in D$ iff there exists a linear function $L : R \to R^{n}$ and a function $η : D_{p} \subseteq R \to R^{n}$ such that:

$δ f_{p} (h) = L (h) + h η (h)$ , for all $h \in D_{p}$
$lim_{h \to 0} η (h) = 0$

Def: Let $g : N \subseteq R \to R^{n}$ and $g^{*} : M \subseteq R \to R^{n}$ be two functions on intervals $N$ and $M$ where $0 \in N \subseteq M$ . We say that $g^{*}$ **********_closely approximates $g$ near $0$ if:

lim_{h \to 0} \frac{g (h) - g^{*} (h)}{h} = 0

for $h \in N$ .

Cor: A function $f : D \subseteq R \to R^{n}$ is differentiable at $p \in D$ iff the difference function $δ_{f, p} : D_{p} \subseteq R \to R^{n}$ can be closely approximated by a linear function near $0$ . If such a close linear approximation exists then it is unique and is $L_{f, p}$

Def: the function $f : D \subseteq R \to R^{n}$ is differentiable at $p \in D$ if there is a linear function $d f_{p} : R \to R^{n}$ (called the differential of $f$ at $p$ ) and a functions $η : D_{p} \subseteq R \to R$ such that:

$f (p + h) - f (p) = d f_{p} (h) + | h | η (h)$ , for all $h \in D_{p}$
$lim_{h \to 0} η (h) = 0$

Other notations for the differential of $f$ at $p$ are: $D_{p} f, D f (p), d f_{p}, d_{p} f$ or $L_{f, p}$

Def: If the function $f : D \subseteq R \to R^{n}$ is differentiable at $p \in D$ , then the $n \times 1$ matrix $J_{f} (p)$ which represents the linear transformation $d f_{p} : R \to R^{n}$ with respect to the standard bases is called the Jacobian of $f$ at $p$ .

The Jacobian of $f$ at $p$ can also be denoted as $J_{f} (p)$ .

Chain Rule: Let $ϕ : E \subseteq R \to R$ and $f : D \subseteq R \to R^{n}$ be differentiable functions and let $ϕ (E) \subseteq D$ . Then $f \circ ϕ : E \subseteq R \to R^{n}$ is differentiable and

d (f \circ ϕ)_{p} = d f_{ϕ (p)} \circ d ϕ_{p}

with the corresponding Jacobian relation

J_{f \circ ϕ} (p) = J_{f} (ϕ (p)) J_{ϕ} (p)