\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

(%i1)

load(draw)$

(%i2)

load(eigen)$

(%i3)

Fx(u, v) := v * cos(u*%pi)$

(%i4)

Fy(u, v) := v * sin(u*%pi)$

(%i5)

Fz(u, v) := u$

(%i6)

vDu(u, v) := [    at(diff(Fx(x, y), x, 1), [x=u,y=v]),    at(diff(Fy(x, y), x, 1), [x=u,y=v]),    at(diff(Fz(x, y), x, 1), [x=u,y=v]) ]$

(%i7)

vDv(u, v) := [    at(diff(Fx(x, y), y, 1), [x=u,y=v]),    at(diff(Fy(x, y), y, 1), [x=u,y=v]),    at(diff(Fz(x, y), y, 1), [x=u,y=v]) ]$

(%i8)

cross (v1, v2) := determinant(matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], v1, v2))$

(%i9)

Normal(u, v) := unitvector(cross(vDu(u, v), vDv(u, v)))$

(%i10)

Light(u, v) := abs(Normal(u, v) . [1, -1, 1])$

(%i11)

wxdraw3d(    enhanced3d=[Light(u, v) * 0.15 + 0.5, u, v],    colorbox=false,    palette=gray,    xu_grid = 100, yv_grid = 100,    parametric_surface(        Fx(u, v), Fy(u, v), Fz(u, v),        u, 0, 3,        v, 0, 1    ) )$

\[\tag{\%{}t11}\label{t11} \]

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