Exercise · ZW-PARAMETRIC-SURFACE-005 · ★★☆ Intermediate
For the cylinder \(\mathbf r(u,v)=\langle\cos u,\,\sin u,\,v\rangle\), the vector \(\mathbf r_u\times\mathbf r_v\) is:
Choose one option.
Show answer
- D. \(\langle\cos u,\,\sin u,\,0\rangle\)
Explanation
With \(\mathbf r_u=\langle-\sin u,\cos u,0\rangle\) and \(\mathbf r_v=\langle0,0,1\rangle\), the cross product is \(\langle\cos u,\sin u,0\rangle\): it points radially outward and has length \(1\).
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-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:
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