Exercise · ZW-PARAMETRIC-SURFACE-008 · ★★★ Advanced
For the graph \(z=x^2-y^2\), parametrized by \(\mathbf r(x,y)=\langle x,\,y,\,x^2-y^2\rangle\), the length of \(\mathbf r_x\times\mathbf r_y\) at \((1,2)\) is:
Choose one option.
A \(\sqrt{6}\) B \(\sqrt{21}\) C \(\sqrt{19}\) D \(21\) Show answer ← Back to the question
Explanation \(\mathbf r_x\times\mathbf r_y=\langle-2x,\,2y,\,1\rangle=\langle-2,4,1\rangle\) and its length is \(\sqrt{4+16+1}=\sqrt{21}\).
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-009 A parametrization of the cone z=√x^2+y^2 is: