CAL1_L15

Derivatives of vector valued functions

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Definition 1(Definition).

Let $\mathbf{f}:(a,b)\to {\mathbb{R}}^n$. Then $\mathbf{f}=(f_1,\dots ,f_n)$ where each component $f_k$ is a real-valued function defined on $(a,b)$. We say that $\mathbf{f}$ is differentiable at each point $c$ in $(a,b)$ if each component $f_k$ is differentiable at $c$ and we define

$${\mathbf{f}}^{\prime }(c)=(f_1^{\prime }(c),\dots ,f_n^{\prime }(c)).$$

Many of the algebraic theorems on differentiation are also valid for vector-valued functions, since they can be applied component-wise. For example, if $\mathbf{f}$ and $\mathbf{g}$ are vector-valued functions differentiable at $c$ and if $\lambda$ is a real valued function differentiable at $c$, then $\mathbf{f}+\mathbf{g}$, $\lambda \mathbf{f}$, and $\mathbf{f}\cdot \mathbf{g}$ are differentiable at $c$ and we have

$$\begin{array}{rl}(\mathbf{f}+\mathbf{g}{)}^{\prime }(c)& ={\mathbf{f}}^{\prime }(c)+{\mathbf{g}}^{\prime }(c),\\ (\lambda \mathbf{f}{)}^{\prime }(c)& ={\lambda }^{\prime }(c)\mathbf{f}(c)+\lambda (c){\mathbf{f}}^{\prime }(c),\\ (\mathbf{f}\cdot \mathbf{g}{)}^{\prime }(c)& ={\mathbf{f}}^{\prime }(c)\cdot \mathbf{g}(c)+\mathbf{f}(c)\cdot {\mathbf{g}}^{\prime }(c).\end{array}$$

There is also a chain rule for differentiating compositions of vector valued functions with real valued functions which is also proved component-wise: If $\mathbf{f}$ is vector valued and $u$ is real valued, if $u^{\prime }(c)$ and ${\mathbf{f}}^{\prime }(u(c))$ exist, and the domain of $\mathbf{f}$ contains a neighborhood of $u(c)$, then the composite function $\mathbf{g}\equiv \mathbf{f}\circ u$ is differentiable at $c$ and

$${\mathbf{g}}^{\prime }(c)={\mathbf{f}}^{\prime }(u(c))u^{\prime }(c).$$

The mean value theorem, as stated here here, does not hold for vector valued functions. A modified version is proved here.

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Partial derivatives

Definition 2(Definition).

Let $S\subseteq {\mathbb{R}}^n$ be open and let $f:S\to \mathbb{R}$. If $\mathbf{x}=(x_1,\dots ,x_n)$ and $\mathbf{c}=(c_1,\dots ,c_n)$ are two points of $S$ having corresponding coordinates equal except for the $k$th, we can consider the limit

$$\underset{x_k\to c_k}{\lim }\frac{f(\mathbf{x})-f(\mathbf{c})}{x_k-c_k}.$$

When this limit exists, it is called the partial derivative of $f$ with respect to the $k$th coordinate, and is denoted by

$$D_{k}f(\mathbf{c}),f_k(\mathbf{c}),\frac{\partial f}{\partial x_k}(\mathbf{c}).$$

Note that a function of $n$ variables can have partial derivatives at a point with respect to each of the variables and yet not be continuous at the point.

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Directional derivatives

Definition 3(Definition).

Let $S\subseteq {\mathbb{R}}^n$ and $\mathbf{f}:S\to {\mathbb{R}}^m$. Let $\mathbf{c}\in S^{\circ }$, and $\mathbf{u}\in {\mathbb{R}}^n$. The directional derivative of $\mathbf{f}$ at $\mathbf{c}$ in the direction $\mathbf{u}$, denoted by ${\mathbf{f}}^{\prime }(\mathbf{c};\mathbf{u})$, is defined by

$${\mathbf{f}}^{\prime }(\mathbf{c};\mathbf{u})=\underset{h\to 0}{\lim }\frac{\mathbf{f}(\mathbf{c}+h\mathbf{u})-\mathbf{f}(\mathbf{c})}{h},$$

whenever the limit on the right exists.

Warning

You can assume $\Vert \mathbf{u}\Vert =1$ if that helps. However, as defined, ${\mathbf{f}}^{\prime }(\mathbf{c};\mathbf{u})$ does depend on $\Vert \mathbf{u}\Vert$. To be precise, ${\mathbf{f}}^{\prime }(\mathbf{c};\lambda \mathbf{u})=\lambda {\mathbf{f}}^{\prime }(\mathbf{c};\mathbf{u})$.

If $\mathbf{u}={\mathbf{e}}_k$, then ${\mathbf{f}}^{\prime }(\mathbf{c};{\mathbf{e}}_k)$ is called a partial derivative and is denoted by $D_k\mathbf{f}(\mathbf{c})$. When $\mathbf{f}$ is real valued this agrees with the definition given in the previous section.

Important

If $\mathbf{F}(t)=\mathbf{f}(\mathbf{c}+t\mathbf{u})$, then ${\mathbf{F}}^{\prime }(t)={\mathbf{f}}^{\prime }(\mathbf{c}+t\mathbf{u};\mathbf{u})$ if either derivative exists.

A function can have a finite directional derivative ${\mathbf{f}}^{\prime }(\mathbf{c};\mathbf{u})$ for every $\mathbf{u}$ but may fail to be continuous at $\mathbf{c}$. The total derivative, a more suitable generalization, does guarantee continuity, and also extends the principal theorems of the one-dimensional derivative.