I am studying geometric measure theory (Herbert Federer) and I have a question about class r homotopies. Here's the definition:

Suppose $U$ is an open subset of $\mathbb{R}^{n}, V$ is an open subset of $\mathbb{R}^{v}$ $f$ and $g$ are functions mapping $U$ into $V . A$ homotopy of class $r$ from $f$ to $g$ is a map $$ h: A \times U \rightarrow V $$ of class $r$ such that $A$ is an open subinterval of $\mathbb{R}, 0 \in A, 1 \in A,$ $$ h(0, x)=f(x) \text { and } h(1, x)=g(x) \text { for } x \in U $$ Whenever $t \in A$ we define $$ h_{t}: U \rightarrow V, \quad h_{t}(x)=h(t, x) \text { for } x \in U $$ hence $h_{0}=f$ and $h_{1}=g ;$ in case $r \geq 1$ we also define $$ h_{t}: U \rightarrow \mathbb{R}^{v}, \quad h_{t}(x)=\langle(1,0), D h(t, x)\rangle \text { for } x \in U $$ hence $\langle(v, w), D h(t, x)\rangle=v h_{t}(x)+\left\langle w, D h_{t}(x)\right\rangle$ for $t \in A, x \in U, v \in \mathbb{R}, w \in \mathbb{R}^{n}$

Now my question is about the last 3 lines:

in case $r \geq 1$ we also define $$ h_{t}: U \rightarrow \mathbb{R}^{v}, \quad h_{t}(x)=\langle(1,0), D h(t, x)\rangle \text { for } x \in U $$ hence $\langle(v, w), D h(t, x)\rangle=v h_{t}(x)+\left\langle w, D h_{t}(x)\right\rangle$ for $t \in A, x \in U, v \in \mathbb{R}, w \in \mathbb{R}^{n}$

Actually, we know that $\langle v, Df(a)\rangle = v \cdot \nabla f(a)$. Also, in the last line it is mentioned that $w \in \mathbb{R}^n$, hence $0 = (0, \cdots,0) \in \mathbb{R}^n$. Now, we have: $$h_t(x)= \langle (1, 0, \cdots, 0), Dh(t,x) \rangle$$ which is equal to $$\begin{align} h_t(x) &= (1, 0, \cdots, 0) \cdot \nabla h(t,x)\\ &= (1, 0 , \cdots, 0) \cdot (\frac{\partial h}{\partial t}, \frac{\partial h}{\partial x_1}, \cdots, \frac{\partial h}{\partial x_n}) \\ &= \frac{\partial h}{\partial t}\end{align} $$

Now, my question is that why it is said that $h_t(x): U \rightarrow \mathbb{R}^{\nu}$? I mean, now we should see that $\frac{\partial h}{\partial t} \in \mathbb{R}^{\nu}$, but I don't get it really! If this is just the differential of $h$, it must lie in $\mathbb{R}^{n-1}$!

Any help is appreciated.