How can we draw a complex 3D graphics using math tools?

Question

I want to draw a 3D graphics, just as follows:

$z=x^{2}+y^{2}, x=0, y=0, z=0, x+y-1=0$ (it's formed by these surfaces)

What math tool should I use and how can I do it with details?
(I've tried intersected lines but failed)

Answer 1

One option is to use CalcPlot3D, which renders this plot:

If, however, you are interested in the portion of the surface $z = x^2 + y^2$ bounded by the planes, then this is given by

\begin{align*} z&=x^2 + y^2\\ x&\ge 0\\ y&\ge 0\\ z&\ge 0\\ x+y-1 &\ge 0 \end{align*}

CalcPlot3D can't handle inequalities, although you can do this in Geogebra using the syntax

z=If(x+y-1>=0&&x>=0&&y>=0&&x^(2)+y^(2)>=0, x^(2)+y^(2))

Note that you do need to express the constraint $z\ge 0$ in terms of $x$ and $y$, which is where the condition $x^2 + y^2 \ge 0$ comes from. This produces the plot:

Answer 2

ContourPlot3D[{z - x^2 - y^2 == 0, x == 0, y == 0, z == 0, 
  x + y - 1 == 0}, {x, -1.2, 1.2}, {y, -1.2, 1.2}, {z, -0.2, 1.2}, 
 ContourStyle -> {Yellow, Magenta, Cyan, Green}, Mesh -> 9]

In Mathematica you can set orthogonal coordinate plot domain limits as desired in ContourPlot3D.