Difference between Path, Curve, Graph and Trace I am having difficulties in understanding the differences between these concepts. We have a new lecturer who loves writing down things in dense mathematical notation (I don't think that's bad but I am just not used to it) and I am not really getting anything. For example, here is how he defined a Path (translation from german):
Let $I\subsetneq \mathbb R$ be any Intervall and let $\vec{x}=I\rightarrow \mathbb R^n$ be any $\mathcal C^k$-map then we call $\vec{x}$ a $\mathcal C^k$-path with values in $\mathbb R$.
What does he mean by $\mathcal C^k$-path?
I hope my translation makes sense. I can't tell because I don't really understand what a path is in the first place. Maybe someone can help me understand these concepts.
Thanks in advance
 A: A path in $\mathbb{R}^n$ is a continuous function $f$ from some interval $[a,b]$ to $\mathbb{R}^n$. We can also not require the interval to be closed. Think of the image of any continuous function on $[a,b]$. Does it not look like you're moving on a path from $f(a)$ to $f(b)$ in the image, without "jumping" across points, as you go from $a$ to $b$ in the domain? 
A $C^k$-path is a path that is differentiable $k$ times and has continuous derivatives on $[a,b]$ each time you differentiate. Since your domain is a subset of $\mathbb{R}$, we just require that each component of your vector-valued function is in $C^k([a,b], \mathbb{R})$.
A graph of a function $f:X\to Y$ is the set of points $(x,y)$ for all $x \in X$.
I need more context to define a trace and a curve, but usually when people say curves, they mean the same thing as the path I just defined.
Note also that two different paths may have the same image. For example, $\gamma_1, \gamma_2: \mathbb{R} \to \mathbb{R}^2$, where $\gamma_1(t) = (\cos(t), \sin(t))$, $\gamma_2(t) = (\cos(2t), \sin(2t))$ both have the same image of a circle in $\mathbb{R}^2$.
A: The idea of a path (to me) is just a $1$-dimensional curve that's parameterized. For example, if we walk around the top half of the unit circle, we can parameterize this path by
\begin{align*}\vec{x}: [0, \pi] &\to \Bbb R^2 \\
t &\mapsto \left(\cos t, \sin t\right).
\end{align*}
I believe the notation $C^k$ just stands for the class of functions that are differentiable $k$-times, and each of these derivatives are continuous. In the example above, since our circular functions have continuous derivatives of all orders, that's an example of a $C^\infty$-path.
I think it's a little odd referring to the function $\vec{x} : I \to \Bbb R^n$ as the path; normally we think of the image $\vec{x}(I) \subseteq \Bbb R^n$ as the path; the actual points in $\Bbb R^n$.
I assume, for traces, you're probably talking about things like this, near the bottom of that link. Essentially, you're focusing on what happens in a particular (hyper)plane. More context would probably be helpful for the rest; they may deserve their own questions.
