Through conformal mapping, the temperature distribution $T$ on a wedge sector can be obtained.
With Mathematica, how to display the distribution?
$$ T(x,y)=\frac{T_0}{\theta_0}\arctan\frac{y}{x}. $$
Obviously, below trial (Cartesian coordinate) is bad, but how to plot the contour in polar coordinate or exactly defined region $\{(r,\theta)|r\le2\,\&\theta\in(0,\frac\pi4)\}$. It seems this post didn't help me out either.
Subscript[T, 0] = 1; Subscript[theta, 0] = Pi/4;
f[x_, y_] := Subscript[T, 0]/Subscript[theta, 0]*ArcTan[y/x]
ContourPlot[f[x, y], {x, 0, Sqrt[2]}, {y, 0, Sqrt[2]}, ContourLabels -> All, PlotRange -> Automatic]
T0 = 1; θ0 = Pi/4; ContourPlot[ T0/θ0 θ, {r, 0, 2}, {θ, 0, π/4}, DisplayFunction -> ReplaceAll[{r_Real, t_Real} :> {r*Cos[t], r*Sin[t]}], PlotPoints -> 80, PlotRange -> All, AspectRatio -> Automatic]mathematica.stackexchange.com/a/300018/72111 $\endgroup$