\documentclass[12pt]{article}
\usepackage{geometry}
\geometry{letterpaper,total={408pt,9.0in}}
%% Custom Page Layout Adjustments (use latex.geometry)
\usepackage{amsmath,amssymb}
\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{positioning}
\usepgfplotslibrary{fillbetween}
\usepackage{pstricks}
\usepackage{phaistos}
\usepackage{xcolor}
\ifdefined\tikzset
\tikzset{ampersand replacement = \amp}
\fi
\newcommand{\order}[1]{\left\lvert#1\right\rvert}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\begin{document}
\pagestyle{empty}
\resizebox{\width}{\height}{
\begin{tikzpicture}
  \begin{axis}[
    compat=1.9,
    x={(-0.2cm,-0.1cm)},
    y={(.375cm,0.0cm)},
    z={(0cm,0.375cm)},
    xlabel={},
    ylabel={},
    zlabel={},
    xmin = -4, xmax = 4,
    ymin = -4, ymax = 4.5,
    zmin = -4, zmax = 4.5,
    xtick = {10},
    ytick = {10},
    ztick = {10},
    samples = 10,
    samples y = 10,
    axis lines = center,
  ]
    \addplot3[surf, shader = faceted interp, domain = -2:0, y domain = -2:0,] ({x+y}, {x^2}, {y^2});
    \addplot3[<->, domain = -2:2, samples y= 0] ({-1+x}, {1^2}, {x^2});
    \addplot3[<->>, domain = -2:2, samples y= 0] ({-1.5+x}, {x^2}, {1.5^2});
    \addplot3[surf, shader = faceted interp, domain = 0:2, y domain = -2:0,] ({x+y}, {x^2}, {y^2});
    \addplot3[<->, domain = -2:2, samples y= 0] ({1.2+x}, {1.2^2}, {x^2});
    \addplot3[surf, shader = faceted interp, domain = -2:0, y domain = 0:2,] ({x+y}, {x^2}, {y^2});
    \addplot3[<->, domain = 0:2, samples y= 0] ({-1+x}, {1^2}, {x^2});
    \addplot3[<->>, domain = -2:2, samples y= 0] ({0.9+x}, {x^2}, {0.9^2});
    \addplot3[surf, shader = faceted interp, domain = 0:2, y domain = -2:0,] ({(x+y)*(abs(x^2-y^2)/(x^2-y^2)+1)/2}, {x^2}, {y^2*(abs(x^2-y^2)/(x^2-y^2)+1)/2 + x^2*(abs(y^2-x^2)/(y^2-x^2)+1)/2});
    \addplot3[->>, domain = 1.5:2, samples y= 0] ({-1.5+x}, {x^2}, {1.5^2});
    \addplot3[surf, shader = faceted interp, domain = 0:2, y domain = 0:2,] ({x+y}, {x^2}, {y^2});
    \addplot3[->>, domain = 0.1:2, samples y= 0] ({0.9+x}, {x^2}, {0.9^2});
    \addplot3[<->>, domain = -2:2, samples y= 0] ({2+x}, {x^2}, {2^2});
    \addplot3[<->>, domain = -2:2, samples y= 0] ({-2+x}, {x^2}, {2^2});
    \addplot3[<->, domain = -2:2, samples y= 0] ({2+x}, {2^2}, {x^2});
    \addplot3[<->, domain = -2:2, samples y= 0] ({-2+x}, {2^2}, {x^2});
    \addplot3[->, domain = 0:2, samples y= 0] ({1.2+x}, {1.2^2}, {x^2});
    \addplot3[mark=none] coordinates {(0, 1, -2)} node {$r(u,v)=(u+v,u^2,v^2)$};
  \end{axis}
\end{tikzpicture}
}
\end{document}
