Partial Directional and Total Derivatives Review

Partial Directional and Total Derivatives Review

We will now review some of the recent material regarding partial, directional, and total derivatives of functions from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $S \subseteq \mathbb{R}^n$ be open, $\mathbf{c} \in S$, and $\mathbf{f} : S \to \mathbb{R}^m$.

• Recall from the Partial Derivatives of Functions from Rn to Rm page that if $\mathbf{e}_k$ denotes the $k^{\mathrm{th}}$ standard basis vector in $\mathbb{R}^n$ then the Partial Derivative of $\mathbf{f}$ at $\mathbf{c}$ with Respect to the $k^{\mathrm{th}}$ Variable is defined to be the following limit provided that it exists:
(1)
\begin{align} \quad D_k \mathbf{f}(\mathbf{c}) = \mathbf{f}'(\mathbf{c}, \mathbf{e}_k) = \lim_{h \to 0} \frac{\mathbf{f}(\mathbf{c} + h\mathbf{e}_k) - \mathbf{f}(\mathbf{c})}{h} \end{align}
• Similarly, recall from the Directional Derivatives of Functions from Rn to Rm page that if $\mathbf{u} \in \mathbb{R}^n$ then the Directional Derivative of $\mathbf{f}$ at $\mathbf{c}$ in the Direction of $\mathbf{u}$ is defined to be the following limit provided that it exists:
(2)
\begin{align} \quad \mathbf{f}' (\mathbf{c}, \mathbf{u}) = \lim_{h \to 0} \frac{\mathbf{f}(\mathbf{c} + h\mathbf{u}) - \mathbf{f}(\mathbf{c})}{h} \end{align}
• We then noted a very simple result. Note that $\mathbf{f} = (f_1, f_2, ..., f_m)$ where for each $k \in \{ 1, 2, ..., m \}$ we have that $f_k : S \to \mathbb{R}$. The definition of the directional derivative above is a limit of a difference of vectors and so the limit exists if and only if the limit in the respective coordinates exist and:
(3)
\begin{align} \quad \mathbf{f}'(\mathbf{c}, \mathbf{u}) = (f_1'(\mathbf{c}, \mathbf{u}), f_2'(\mathbf{c},\mathbf{u}), ..., f_m'(\mathbf{c},\mathbf{u})) \end{align}
• On the Directional Derivatives of Functions from Rn to Rm and Continuity page we noted a very important fact. The existence of all directional derivatives of $\mathbf{f}$ at a point $\mathbf{c}$ does NOT imply the continuity of $\mathbf{f}$ at $\mathbf{c}$. We used the following example $f : \mathbb{R}^2 \to \mathbb{R}$ to prove this:
(4)
\begin{align} \quad f(x, y) = \quad f(x, y) = \left\{\begin{matrix} \frac{x^2y}{(x^4 + y^2)} & \mathrm{if} \: (x, y) \neq (0, 0) \\ 0 & \mathrm{if} \: (x, y) = (0, 0) \end{matrix}\right. \end{align}
• We showed that the directional derivatives of $f$ at $\mathbf{c} = (0, 0)$ exist for all $\mathbf{u} \in \mathbb{R}^2$ but that $f$ is actually discontinuous at $\mathbf{c}$.
• On the Differentiability and the Total Derivative of Functions from Rn to Rm we looked at a more satisfactory definition for differentiation. We said that $\mathbf{f}$ is Differentiable at $\mathbf{c}$ if there exists a linear function $\mathbf{T}_{\mathbf{c}} : \mathbb{R}^n \to \mathbb{R}^m$ which is called the Total Derivative of $\mathbf{f}$ at $\mathbf{c}$ such that:
(5)
\begin{align} \quad \mathbf{f} (\mathbf{c} + \mathbf{v}) = \mathbf{f}(\mathbf{c}) + \mathbf{T}_{\mathbf{c}}(\mathbf{v}) + \| \mathbf{v} \| \mathbf{E}_{\mathbf{c}} (\mathbf{v}) \end{align}
• Where $\mathbf{E}_{\mathbf{c}}(\mathbf{v}) \to \mathbf{0}$ as $\mathbf{v} \to \mathbf{0}$. The notation "$\mathbf{T}_{\mathbf{c}}$" is sometimes replaced by "$\mathbf{f}'(\mathbf{c})(\mathbf{v})$".
• We noted that if $f$ is a single-variable real-valued function then $f$ being differentiable in the familiar sense is indeed equivalent to $f$ being differentiable from the definition above. If $f$ is differentiable at $c$ then $f'(c)$ exists and the total derivative of $f$ at $c$ is the linear function $T_c(v) = f'(c) \cdot v$. Since $f'(c)$ is constant, $T_c(v)$ is simply a line which is of course a linear function.
• On the Differentiable Functions from Rn to Rm and Their Total Derivatives page we looked a a very important theorem which said that if $\mathbf{f}$ is differentiable at $\mathbf{c}$ then all of the directional derivatives of $\mathbf{f}$ at $\mathbf{c}$ exist and moreover, the partial derivative of $\mathbf{f}$ at $\mathbf{c}$ in the direction of $\mathbf{u}$ can be obtained by plugging in $\mathbf{u}$ to the total derivative $\mathbf{T}_{\mathbf{c}}$ of $\mathbf{f}$ at $\mathbf{c}$, i.e.:
(6)
\begin{align} \quad \mathbf{T}_{\mathbf{c}}(\mathbf{u}) = \mathbf{f}'(\mathbf{c}, \mathbf{u}) \end{align}
(7)
\begin{align} \quad \mathbf{T}_{\mathbf{c}}(\mathbf{v}) = v_1\mathbf{f}'(\mathbf{c}, \mathbf{e}_1) + v_2\mathbf{f}'(\mathbf{c}, \mathbf{e}_2) + ... + v_n\mathbf{f}'(\mathbf{c}, \mathbf{e}_n) = \sum_{k=1}^{n} v_k \mathbf{f}'(\mathbf{c}, \mathbf{e}_k) \end{align}
(8)
\begin{align} \quad \mathbf{T}_{\mathbf{c}}(\mathbf{v}) = v_1D_1 \mathbf{f}(\mathbf{c}) + v_2D_2 \mathbf{f}(\mathbf{c}) + ... + v_n D_n\mathbf{f}(\mathbf{c}) = \sum_{k=1}^{n} v_k D_k \mathbf{f}(\mathbf{c}) \end{align}