A map from an open set $\mathcal{D} \subset \mathbb{R}^n$ to $\mathbb{R}^m$, denoted $f : \mathcal{D} \subset \mathbb{R}^n \longrightarrow \mathbb{R}^m$, is a rule that assigns each $x \in \mathcal{D}$ a value $f(x) \in \mathbb{R}^m$. Decomposing $f(x)$ into components, we can write such a map as $$f(x) = \left( f_1(x), f_2(x), \dots, f_m(x) \right), \qquad x = \left( x_1, x_2, \dots, x_n \right) \in \mathcal{D} \subset \mathbb{R}^n, $$ where each component $f_i(x), i=1, 2, \dots, m$, defines a function on $\mathcal{D} \subset \mathbb{R}^n$. We say that $f(x)$ is differentiable on $\mathcal{D}$ if each one of the component functions $f_i(x), i=1, 2, \dots, m$, are differentiable on $\mathcal{D}$.
Suppose that $f : \mathcal{D} \subset \mathbb{R}^n \longrightarrow \mathbb{R}^m$ is differentiable map on $\mathcal{D}$ and $$r(t) = \left( r_1(t), r_2(t), \dots, r_n(t) \right), \qquad -1\lt t \lt 1,$$ is a differentiable curve on $\mathbb{R}^n$ that lies in $\mathcal{D}$. We can use the map $f(x)$ to define a differentiable curve on $\mathbb{R}^m$ by setting $$\gamma(t) = f(r(t)),$$ or equivalently, $$ \gamma(t) = \left( \gamma_1(t), \gamma_2(t), \dots, \gamma_m(t) \right) $$ where $$\gamma_i(t) = f_i \left( r_1(t), r_2(t), \dots, r_n(t) \right), \qquad i = \left( 1, 2, \dots, m \right). $$ We then know the vectors $$v = r'(0) \quad \text{and} \quad w = \gamma'(0)$$ area tangent to the curves $r(t)$ and $\gamma(t)$ at the points $x_0 = r(0) \in \mathcal{D} \subset \mathbb{R}^n$ and $\gamma_0 \in \mathbb{R}^m$, respectively. Using column notating to denote the tangent vectors $v$ and $w$, that is, $$ v = \left(\begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_n \end{array}\right) \quad \text{and} \quad w = \left(\begin{array}{c} w_1 \\ w_2 \\ \vdots \\ w_n \end{array}\right), $$ show that vectors $v$ and $w$ are related by the linear relation $$ w = Df(x_0)v, $$ where $Df(x)$, for $x \in \mathcal{D} \subset \mathbb{R}^n$, is the matrix defined by $$ Df(x) = \left(\begin{array}{ccccc} \dfrac{\partial f_1}{\partial x_1}(x) & \dfrac{\partial f_1}{\partial x_2}(x) & \dfrac{\partial f_1}{\partial x_3}(x) &\cdots & \dfrac{\partial f_1}{\partial x_n}(x) \\ \dfrac{\partial f_2}{\partial x_1}(x) & \dfrac{\partial f_2}{\partial x_2}(x) & \dfrac{\partial f_2}{\partial x_3}(x) &\cdots & \dfrac{\partial f_2}{\partial x_n}(x) \\ \dfrac{\partial f_3}{\partial x_1}(x) & \dfrac{\partial f_3}{\partial x_2}(x) & \dfrac{\partial f_3}{\partial x_3}(x) &\cdots & \dfrac{\partial f_3}{\partial x_n}(x) \\ \vdots & \vdots & \vdots & & \vdots \\ \dfrac{\partial f_m}{\partial x_1}(x) & \dfrac{\partial f_m}{\partial x_2}(x) & \dfrac{\partial f_m}{\partial x_3}(x) &\cdots & \dfrac{\partial f_m}{\partial x_n}(x) \end{array}\right).$$ [Hint: Use the chain rule.]
I understand the individual concepts in this problem, but I can't get started on actually answering it. How would one go about putting the pieces together here? Sorry that I haven't done any working, but I need something to get started with.
Sometimes when you have a vector equation and you're not sure how to proceed, but you know you need to apply the chain rule, the best thing to do is to write a single component of the vector equation.
Consider say $w_1$ and recall that matrix multiplication $w = Av$ is written in components as $w_i = \sum\limits_{j} A_{ij}v_j$:
\begin{align} w_1 &= \sum\limits_{j=1}^n (Df(x_0))_{1j}v_j\\ &= \sum\limits_{j=1}^n \dfrac{\partial f_1}{\partial x_j}(x_0)r_j'(0)\\ &= \frac{d}{dt}\gamma_1(0) \end{align}
Can you take it from here?