Exercise · ZW-PARAMETRIC-SURFACE-007 · ★★☆ Intermediate
A parametrization of the sphere of radius \(2\) centered at the origin is:
Choose one option.
Show answer
- C. \(\langle2\sin\varphi\cos\theta,\,2\sin\varphi\sin\theta,\,2\cos\varphi\rangle\)
Explanation
Only the parametrization with \(2\sin\varphi\cos\theta,\,2\sin\varphi\sin\theta,\,2\cos\varphi\) satisfies \(x^2+y^2+z^2=4\); the others do not describe a sphere of radius \(2\).
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-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…
- ZW-PARAMETRIC-SURFACE-009 A parametrization of the cone z=√x^2+y^2 is: