In the first three exercises, find an equation for the plane tangent to the given surface at the specified point.
\(x=2u,\qquad y=u^2+v,\qquad z=v^2,\qquad at\, (0,1,1)\)
\(x=u^2-v^2,\qquad y=u+v,\qquad z=u^2+4v, at\, (-\frac{1}{4},\frac{1}{2},2)\)
\(x=u^2,\qquad y=u\sin e^v,\qquad z=\frac{1}{3}u\cos e^v, at\, (13,-2,1)\)
At what points are the surfaces in Exercises 1 and 2 regular?
In the next two exercises, find all points \((u_{0},v_{0})\), where \(S = \Phi(u_{0},v_{0})\) is not smooth (regular).
\(\Phi(u,v) = (u^2 - v^2, u^2 + v^2, v)\)
\(\Phi(u,v) = (u - v, u + v, 2uv)\)
Match the following parameterizations to the surfaces shown in the figures.
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Match the following parametrizations to the surfaces shown in the figures.
Find an expression for a unit vector normal to the surface \[ x=\cos v\sin u,\qquad y=\sin v \sin u,\qquad z=\cos u \] at the image of a point \((u,v)\) for \(u\) in \([0,\pi]\) and \(v\) in \([0,2\pi]\). Identify this surface.
Repeat the previous exercise (the ninth) for the surface \[ x=3\cos \theta\sin \phi,\qquad y=2\sin \theta\sin \phi,\qquad z=\cos \phi \] for \(\theta\) in \([0, 2\pi]\) and \(\phi\) in \([0,\pi]\).
Repeat the ninth exercise for the surface \[ x=\sin v,\qquad y=u,\qquad z=\cos v \] for \(0\leq v\leq 2\pi\) and \(-1\leq u\leq 3\).
Repeat the ninth exercise for the surface \[ x=(2-\cos v)\cos u,\quad y=(2-\cos v)\sin u,\quad z=\sin v \] for \(-\pi \leq u\leq \pi,-\pi \leq v\leq \pi\). Is this surface regular?
Find the equation of the plane tangent to the surface \(x=u^2,y=v^2,z=u^2+v^2\) at the point \(u=1,v=1\).
Find a parametrization of the surface \(z=3x^2+8{\it xy}\) and use it to find the tangent plane at \(x=1,y=0,z=3\). Compare your answer with that using graphs.
Find a parametrization of the surface \(x^3\,{+}\,3{\it xy} \,{+}\,z^2\,{=}\,2, z>0\), and use it to find the tangent plane at the point \(x\,{=}\,1,y\,{=}\,1/3,z\,{=}\,0\). Compare your answer with that using level sets.
Consider the surface in \({\mathbb R}^3\) parametrized by \begin{eqnarray*} &&{\Phi}(r,\theta)=(r\cos \theta,r\sin \theta,\theta),\quad 0\leq r\leq 1\\ && \hbox{and}\quad 0\leq \theta \leq 4\pi. \end{eqnarray*}
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Given a sphere of radius 2 centered at the origin, find the equation for the plane that is tangent to it at the point \((1,1,{\textstyle\sqrt{2})}\) by considering the sphere as:
A parametrized surface is described by a differentiable function \({\Phi}\colon\, {\mathbb R}^2\to {\mathbb R}^3\). According to Chapter 13, the derivative should give a linear approximation that yields a representation of the tangent plane. This exercise demonstrates that this is indeed the case.
Consider the surfaces \({\Phi}_1(u,v)=(u,v,0)\) and \({\Phi}_2(u,v)=(u^3,v^3,0)\).
The image of the parametrization \begin{eqnarray*} \Phi(u,v) & = &( x(u,v),y(u,v),z(u,v) ) \\[3pt] & = &( a \sin u \cos v, b \sin u \sin v, c \cos u )\\[-16pt] \end{eqnarray*} with \(b<a\), \(0 \leq u \leq \pi\), \(0 \leq v \leq 2\pi\) parametrizes an ellipsoid.
The image of the parametrization \begin{eqnarray*} \Phi(u,v) & = &( x(u,v),y(u,v),z(u,v) ) \\[3pt] & = &( (R + r \cos u) \cos v, (R + r \cos u) \sin v, r \sin u ) \end{eqnarray*} with \(0 \leq u, v \leq 2\pi\), \(0< r <1\) parametrizes a torus (or doughnut) \(S\).
Let \({\Phi}\) be a regular surface at \((u_0,\!v_0)\); that is, \({\Phi}\) is of class \(C^1\) and \({\bf T}_u\!\times\!{\bf T}_v\!\neq\! {\bf 0}\) at \((u_0,\!v_0)\).