Exercise · ZW-PARAMETRIC-SURFACE-009 · ★★★ Advanced
A parametrization of the cone \(z=\sqrt{x^2+y^2}\) is:
Choose one option.
Show answer
- D. \(\langle v\cos u,\,v\sin u,\,v\rangle\), with \(v\ge0\)
Explanation
With \(x=v\cos u\), \(y=v\sin u\) we get \(\sqrt{x^2+y^2}=v=z\). The others give a paraboloid, a cylinder and a flat disk.
Review the concept: Parametric surface
A parametric surface is the set of points in space described by a vector function \(\mathbf r\) of two parameters \(u\) and \(v\) as they range over a region \(D\) of the plane.
\[\mathbf r(u,v)=\langle x(u,v),\,y(u,v),\,z(u,v)\rangle,\quad (u,v)\in D\]
More practice problems: parametric surface
- ZW-PARAMETRIC-SURFACE-001 The surface mathbf r(u,v)=langle u, v, u^2+v^2rangle is:
- ZW-PARAMETRIC-SURFACE-002 For mathbf r(u,v)=langle u^2v, u+v, v^3rangle, the vector mathbf r_u at (u,v)=(1,2) is:
- ZW-PARAMETRIC-SURFACE-003 For the same surface mathbf r(u,v)=langle u^2v, u+v, v^3rangle, the vector mathbf r_v at (1,2) is:
- ZW-PARAMETRIC-SURFACE-004 A normal vector to mathbf r(u,v)=langle u, v, u^2+v^2rangle at (u,v)=(1,1) is:
- ZW-PARAMETRIC-SURFACE-005 For the cylinder mathbf r(u,v)=langlecos u, sin u, vrangle, the vector mathbf r_u×mathbf r_v is:
- ZW-PARAMETRIC-SURFACE-006 The tangent plane to mathbf r(u,v)=langle u, v, u^2+v^2rangle at the point (1,1,2) is:
- ZW-PARAMETRIC-SURFACE-007 A parametrization of the sphere of radius 2 centered at the origin is:
- ZW-PARAMETRIC-SURFACE-008 For the graph z=x^2-y^2, parametrized by mathbf r(x,y)=langle x, y, x^2-y^2rangle, the length of ma…