4.1.E: Geometry, Limits, and Continuity (Exercises)
- Page ID
- 78229
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
For each of the following, plot the surface parametrized by the given function.
(a) \(f(s, t)=\left(t^{2} \cos (s), t^{2} \sin (s), t^{2}\right), 0 \leq s \leq 2 \pi, 0 \leq t \leq 3\)
(b) \(f(u, v)=(3 \cos (u) \sin (v), \sin (u) \sin (v), 2 \cos (v)), 0 \leq u \leq 2 \pi, 0 \leq v \leq \pi\)
(c) \(g(s, t)=((4+2 \cos (t)) \cos (s),(4+2 \cos (t)) \sin (s), 2 \sin (t)), 0 \leq s \leq 2 \pi, 0 \leq t \leq 2 \pi\)
(d) \(f(s, t)=((5+2 \cos (t)) \cos (s), 2(5+2 \cos (t)) \sin (s), \sin (t)), 0 \leq s \leq 2 \pi, 0 \leq t \leq 2 \pi\)
(e) \(h(u, v)=(\sin (v),(3+\cos (v)) \cos (u),(3+\cos (v)) \sin (u)), 0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\)
(f) \(g(s, t)=\left(s, s^{2}+t^{2}, t\right),-2 \leq s \leq 2,-2 \leq t \leq 2\)
(g) \(f(x, y)=(y \cos (x), y, y \sin (x)), 0 \leq x \leq 2 \pi,-5 \leq y \leq 5\)
Exercise \(\PageIndex{2}\)
Suppose \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) and we define \(F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) by \(F(s, t)=(s, t, f(s, t))\). Describe the surface parametrized by \(F\).
- Answer
-
The surface is the graph of \(f\).
Exercise \(\PageIndex{3}\)
Find a parametrization for the surface that is the graph of the function \(f(x,y)=x^{2}+y^{2}\).
- Answer
-
\(F(s, t)=\left(s, t, s^{2}+t^{2}\right)\)
Exercise \(\PageIndex{4}\)
Make plots like those in Figure 4.1.4 for each of the following vector fields. Experiment with the rectangle used for the grid, as well as with the number of vectors drawn.
(a) \(f(x, y)=(y,-x)\)
(b) \(g(x, y)=(y,-\sin (x))\)
(c) \(f(u, v)=\left(v, u-u^{3}-v\right)\)
(d) \(f(x, y)=\left(x\left(1-y^{2}\right)-y, x\right)\)
(e) \(f(x, y, z)=\left(10(y-x), 28 x-y-x z,-\frac{8}{3} z+x y\right)\)
(f) \(f(x, y, z)=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}(x, y, z)\)
(g) \(g(u, v, w)=-\frac{1}{(u-1)^{2}+(v-2)^{2}+(w-1)^{2}}(u-1, v-2, w-1)\)
Exercise \(\PageIndex{5}\)
Find the set of points in \(\mathbb{R}^2\) for which the vector field
\[ f(x, y)=\left(4 x \sin (x-y), \frac{4 x+3 y}{2 x-y}\right) \nonumber \]
is continuous.
Exercise \(\PageIndex{6}\)
For which points in \(\mathbb{R}^n\) is the vector field
\[ f(\mathbf{x})=\frac{\mathbf{x}}{\log (\|\mathbf{x}\|)} \nonumber \]
a continuous function?
- Answer
-
\(\left\{\mathbf{x}: \mathbf{x} \in \mathbb{R}^{n}, \mathbf{x} \neq \mathbf{0}\right\}\)