Exercise · ZW-PARAMETRIC-SURFACE-004 · ★★☆ Intermediate
A normal vector to \(\mathbf r(u,v)=\langle u,\,v,\,u^2+v^2\rangle\) at \((u,v)=(1,1)\) is:
Choose one option.
Show answer
- B. \(\langle -2,-2,1\rangle\)
Explanation
\(\mathbf r_u=\langle1,0,2u\rangle\), \(\mathbf r_v=\langle0,1,2v\rangle\) and \(\mathbf r_u\times\mathbf r_v=\langle-2u,-2v,1\rangle\), which at \((1,1)\) gives \(\langle-2,-2,1\rangle\).
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-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…
- ZW-PARAMETRIC-SURFACE-009 A parametrization of the cone z=√x^2+y^2 is: