Abuse of notation or am I missing something I'm reading a text carefully and I've come across a part that's somewhat confusing. 
Suppose there exists a $C^1$ function $F: \mathbb{R}^n \to \mathbb{R}^m$ and by fixing $x$ and $y$ we define $\phi(t) = F(x+ty)$. My text says that by the fundamental theorem of calculus we can write $$F(x+y) = F(x) + \int_0^1 DF(x+ty)y \ dt$$
How is the operation in the integrand defined if $DF$ maps to something $m$-dimensional and $y$ is $n$-dimensional. I'm being a bit of a stickler but I want to make sure there are no gaps in my understanding. 
 A: For each value $x + ty$, the derivative $DF(x + ty)$ is itself a linear map $\mathbb R^n \to \mathbb R^m$. Hence acting on $y \in \mathbb R^n$,
$$DF(x + ty)y \in \mathbb R^m$$
A: In fact, the derivative of a differentiable map $F: \mathbb{R}^{n} \to \mathbb{R}^{m}$ at a point $x \in \mathbb{R}^{n}$, I denote which by $dF^{x}$, is by definition a linear map $\mathbb{R}^{n} \to \mathbb{R}^{m}$. A reason that many authors write $DF(x)$ instead of $dF^{x}$ is: a map $f: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is linear iff there is some $m \times n$ matrix $A$ such that $f(x) = Ax$ for all $x \in \mathbb{R}^{n}$; the Jacobian matrix of $f$ at a point $x \in \mathbb{R}^{n}$ is by definition the matrix of $df^{x}$ with respect to the usual bases and is denoted by $Df(x)$; therefore $df^{x}(y) = Df(x)\cdot y$ for all $y \in \mathbb{R}^{n}$.
By chain rule, formally we have $\varphi'(t) = DF(x+ty)y$; hence the map being integrated is in fact the map $t \mapsto DF(x+ty)y: [0,1] \to \mathbb{R}^{m}$, but the map $t \mapsto DF(x+ty)$ carries $[0,1]$ to $\mathscr{C}(\mathbb{R}^{n}, \mathbb{R}^{m})$.
I prefer not to use $Df(x)$ to denote the derivative of $f$ at $x$; I reserve it for Jacobian matrix. You are encouraged to check whether the author use $Df(x)$ to mean a map or a matrix; if the former is the case and if you are not yet familiar with that, then you can write $Df(x)(y)$ to remind yourself.
