Suppose I have a function
$$f(x,y) = xy$$
or more complicated: $$f(x,y) = x^Ty$$
Suppose I set my $y$ to be some constant (vector) $c$.
What are the resulting expressions
$$f(x,y) = xc$$
$$f(x,y) = x^Tc$$ called?
(Why would someone be interested in the name? The reason is because sometimes I would like to graph a $\mathbb{R}^2\times \mathbb{R}^2$ function by fixing one of its arguments, so it is a 3D function. But I am not sure how to refer to the resulting expression.)
I know it is not level (surface) set. Is it the projection of a function onto $D$ where $x \in D$? I' not sure.
Let $f:D \rightarrow \mathbb{R}$ be a scalar-valued function, where $D\subseteq \mathbb{R}^2$ is a domain. Let's write the coordinates of $D$ as $(x,y)$. Let $c$ and $d$ be constants. Then $\{ (x,d)\in D \}$ and $\{(c,y)\in D \}$ are called $\textbf{grid lines}$ (which may be composed of multiple line segments if $D$ is not convex), while $\{ (x,d,f(x,d))\in \mathbb{R}^3: (x,d)\in D \} $ and $\{ (c,y,f(c,y))\in \mathbb{R}^3: (c,y)\in D \} $ are called $\textbf{grid curves}$.
In your example, if $f:\mathbb{R}^2\times \mathbb{R}^2\rightarrow \mathbb{R}$, then the graph of $f(x,c)$ could be called a $\textbf{grid surface}$ (since $x$ is a parameter in a $2$-dimensional space).
You can define higher-dimensional analogues by calling them $\textbf{grid solids}$, $\textbf{grid hypersurfaces}$, $\textbf{hypersurfaces}$, etc.