Cylinder

ParametricPlot3D[{Cos[u], Sin[u], v}, {u, 0, 2  π}, {v, -1, 
        1}]

ParametricPlot3D[{Cos[u], Sin[u], v}, {u, 0, π/2}, {v, -1, 
        1}]

Plane

r  =  {1, -2, 0} + u {3, 1, 1} + v {0, 4, -1}

ParametricPlot3D[ {1 + 3 u, -2 + u + 4 v, u - v}, {u, -1, 
        1}, {v, -1, 1}]

Sphere

ParametricPlot3D[{3 Sin[u] Cos[v], 3  Sin[u] Sin[v], 3  Cos[u]}, {u, 
        0, π}, {v, 0, 2  π}]

Plot3D[(Sqrt[9  -  (x^2) - (y^2)]), {x, -3, 3}, {y, -3, 
        3}]

Cone

ParametricPlot3D[{v Cos[u]/2, v Sin[u]/3,   v}, {v, -3, 3}, {u, 
        0, 2  π}]

ParametricPlot3D[{x, y, (Sqrt[4  (x^2) + 9  (y^2)])}, {x, -4, 
        4}, {y, -3, 3}]

Paraboloid

ParametricPlot3D[{v Cos[u], v Sin[u],   (v^2)}, {v, 0, 3}, {u, 0, 
        2  π}]

ParametricPlot3D[{x, y, (x^2) + (y^2)}, {x, -3, 3}, {y, -3, 
        3}]

ParametricPlot3D[{v Cos[u],  (v^2), v Sin[u]}, {v, 0, 3}, {u, 0, 
        2  π}]

Surface of Revolution

y  =  (x^3) revolved about the x-axis.

ParametricPlot3D[{x,  (x^3) Cos[u],  (x^3) Sin[u]}, {x, -1.5, 
        1.5},  {u, 0, 2  π}]

Tangent Plane

surface = 
      ParametricPlot3D[{u v,  u  +  (v^2),  v  -  (u^2)},  {u, -3, 
          3}, {v, -3, 3}, Boxed -> False, Axes -> False,  
        ViewPoint -> {2, 3, 1}]

Tangent plane at (1, 2, 0).

r  =  {u v,  u  +  (v^2),  v  -  (u^2)};

p = r  /.  {u -> 1,  v -> 1}

a  =  D[r, u]  /.  {u -> 1,  v -> 1}

b  =  D[r, v]  /.  {u -> 1,  v -> 1}

tplane  =  
      ParametricPlot3D[
        p + t a  +  s b,  {t, -1, 1},  {s, -1, 1}]

Show[surface, tplane, ViewPoint -> {3, 2, 1}]

Surface Area of a Torus

a = 1; b = 2;
ParametricPlot3D[{(b + a Sin[v]) Cos[u], (b + a Sin[v]) 
          Sin[u], a Cos[v]}, {v, 0, 2  π}, {u, 0, 2  π}]

a = 1; b = 4;
ParametricPlot3D[{(b + a Sin[v]) Cos[u], (b + a Sin[v]) 
          Sin[u], a Cos[v]}, {v, 0, 2  π}, {u, 0, 2  π}]

a = 1; b = 4;
ParametricPlot3D[{(b + a Sin[v]) Cos[u], (b + a Sin[v]) 
          Sin[u], a Cos[v]}, {v, 0, 2  π}, {u, 0, 2  π}]

a = 1; b = 1.25;
ParametricPlot3D[{(b + a Sin[v]) Cos[u], (b + a Sin[v]) 
          Sin[u], a Cos[v]}, {v, 0, 2  π}, {u, 0, 2  π}]

Clear[a, b]

r  = {(b + a Cos[v]) Cos[u], (b + a Cos[v]) Sin[u], 
          a Sin[v]};

c  =  Cross[D[r, v], D[r, u]] // Simplify

Sqrt[c . c] // Simplify

Integrate[
      a (b + a Cos[v]), {u, 0, 2  π}, {v, 0, 2  π}]

Trefoil Knot

Clear[t]

trefoil = {Sin[3  t], Sin[t] + 2  Sin[2  t], 
          Cos[t] - 2  Cos[2  t]};

ParametricPlot3D[trefoil, {t, 0, 2  π}]

TubePlotFrenet[curve_List, {var_, min_, max_}, radius_, opts___] := 
      Module[{tangent, unitTangent, normal, unitNormal, biNormal}, 
        tangent = D[curve, t]; 
unitTangent = 
          tangent/Sqrt[tangent . tangent]; 
normal = 
          D[unitTangent, t]; 
unitNormal = 
          normal/Sqrt[normal . normal]; 
biNormal = 
          Cross[unitTangent, unitNormal]; 
ParametricPlot3D[
          curve + radius Cos[s] unitNormal + radius Sin[s] biNormal // 
            Evaluate, {var, min, max}, {s, 0, 2  π}, opts]]

TubePlotFrenet[trefoil, {t, 0, 2  π}, 0.3, Axes -> None, 
      Boxed -> False, ViewPoint -> {10, 0, 0}, 
      PlotPoints -> {64, 16}]

www.unca.edu/~mcmcclur/java/LiveMathematica/trefoil.html