Differentiabilty of vector valued functions of R

Subjects: Vector Analysis
Links: Limits and Continuity of Vector Valued Functions of R, Differentiability of Vector valued functions of Rn

Differentiability

Def: Let $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable on $x_0\in D$ if the limit exists:

then its value is the derivative of $f$ on $x_0$, usually denoted as $f^{\prime }(x_0)$.

Th: Let $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable on $x_0\in D$, iff for each $1\le i\le n$ component functions $f_i$ is differentiable at $x_0$. If $f$ is differentiable at $x_0$, then:

Def: Let $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable on $x_0\in D$

• The derivative $f^{\prime }(x_0)$ will also be called the tangent vector of $f$ at $x_0$
• If $f^{\prime }(x_0)\ne 0$, then the straight line in ${\mathbb{R}}^n$ through $f(x_0)$ with direction $f^{\prime }(x_0)$ will be called the ****tangent line of $f$ at $x_0.$ It is the set $\{f(x_0)+t{f}^{\prime }(x_0)\mid t\in \mathbb{R}\}$

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

Th: Let $f,g:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\varphi :D\subseteq \mathbb{R}\to \mathbb{R}$ be differentiable at $x_0\in D.$ Then:

• $f+g$ is differentiable at $x_0$, and $(f+g{)}^{\prime }(x_0)=f^{\prime }(x_0)+g^{\prime }(x_0)$
• $f\cdot g$ is differentiable at $x_0$, and $(f\cdot g{)}^{\prime }(x_0)=f^{\prime }(x_0)\cdot g(x_0)+f(x_0)\cdot g^{\prime }(x_0)$ Alternatively it can be expressed using the inner product notation: ${⟨f,g⟩}^{\prime }(x_0)=⟨f^{\prime }(x_0),g(x_0)⟩+⟨f(x_0),g^{\prime }(x_0)⟩$
• $\varphi f$ is differentiable at $x_0$, and $(\varphi f{)}^{\prime }(x_0)=\varphi (x_0)f^{\prime }(x_0)+{\varphi }^{\prime }(x_0)f(x_0)$
• If $n=3$, then $f\times g$ is differentiable at $x_0$, and $(f\cdot g{)}^{\prime }(x_0)=f^{\prime }(x_0)\cdot g(x_0)+f(x_0)\cdot g^{\prime }(x_0)$

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

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

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

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

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

In the case that $\varphi$ is strictly increasing, then we say that $f$ and $h$ are properly equivalent. In the other hand, if $\varphi$ 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 ${\mathcal{C}}^{\mathcal{1}}$ parametrization of $C$.

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

Th: Let $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $h:E\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be equivalent ${\mathcal{C}}^{\mathcal{1}}$ parametrizations of a curve $C$ in ${\mathbb{R}}^n$ related by a differentiable function $\varphi :E\to D$. Let $h=f\circ \varphi$ and $\varphi$ be strictly monotonic, then whenever $\varphi (u)=t$ 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 $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is said to be smooth/regular if it is ${\mathcal{C}}^{\mathcal{1}}$ and if $f^{\prime }(t)\ne 0$ for all $t\in D$

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

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

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

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

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

Def: A function $f:[a,b]\to {\mathbb{R}}^n$ is said to be piecewise ${\mathcal{C}}^{\mathcal{1}}$ (piecewise smooth) if there is a partition $\mathcal{P}\mathcal{=}\{{\mathcal{x}}_{\mathcal{i}}{\}}_{\mathcal{i}\mathcal{=}\mathcal{0}}^{\mathcal{m}}$ of $[a,b]$ such that for each $1\le k\le m$ the function $f$ is restricted to the subinterval $[x_{k-1},x_k]$ is ${\mathcal{C}}^{\mathcal{1}}$ (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 $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$, and $p\in D$. Let’s consider the function $\delta f_p:D_p\subseteq \mathbb{R}\to {\mathbb{R}}^n$ where $D_p=\{h\in \mathbb{R}\mid p+h\in D\}$,

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

• $\delta f_p(h)=L(h)+h\eta (h)$, for all $h\in D_p$
• $\underset{h\to 0}{lim}\eta (h)=\mathbf{0}$

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

for $h\in N$.

Cor: A function $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable at $p\in D$ iff the difference function ${\delta }_{f,p}:D_p\subseteq \mathbb{R}\to {\mathbb{R}}^n$ 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 $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable at $p\in D$ if there is a linear function $d{f}_p:\mathbb{R}\to {\mathbb{R}}^n$ (called the differential of $f$ at $p$) and a functions $\eta :D_p\subseteq \mathbb{R}\to \mathbb{R}$ such that:

• $f(p+h)-f(p)=d{f}_p(h)+|h|\eta (h)$, for all $h\in D_p$
• $\underset{h\to 0}{lim}\eta (h)=\mathbf{0}$

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

Def: If the function $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable at $p\in D$, then the $n\times 1$ matrix $J_f(p)$ which represents the linear transformation $d{f}_p:\mathbb{R}\to {\mathbb{R}}^n$ 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 $\varphi :E\subseteq \mathbb{R}\to \mathbb{R}$ and $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be differentiable functions and let $\varphi (E)\subseteq D$. Then $f\circ \varphi :E\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is differentiable and

with the corresponding Jacobian relation