4.1.E: Geometría, Límites y Continuidad (Ejercicios)
- Page ID
- 111696
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Ejercicio\(\PageIndex{1}\)
Para cada una de las siguientes, trazar la superficie parametrizada por la función dada.
(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)\(f(x, y)=(y \cos (x), y, y \sin (x)), 0 \leq x \leq 2 \pi,-5 \leq y \leq 5\)
Ejercicio\(\PageIndex{2}\)
Supongamos\(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) y definimos\(F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) por\(F(s, t)=(s, t, f(s, t))\). Describir la superficie parametrizada por\(F\).
- Contestar
-
La superficie es la gráfica de\(f\).
Ejercicio\(\PageIndex{3}\)
Encuentra una parametrización para la superficie que es la gráfica de la función\(f(x,y)=x^{2}+y^{2}\).
- Contestar
-
\(F(s, t)=\left(s, t, s^{2}+t^{2}\right)\)
Ejercicio\(\PageIndex{4}\)
Hacer gráficas como las de la Figura 4.1.4 para cada uno de los siguientes campos vectoriales. Experimenta con el rectángulo utilizado para la cuadrícula, así como con el número de vectores dibujados.
(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)\(g(u, v, w)=-\frac{1}{(u-1)^{2}+(v-2)^{2}+(w-1)^{2}}(u-1, v-2, w-1)\)
Ejercicio\(\PageIndex{5}\)
Encuentra el conjunto de puntos en\(\mathbb{R}^2\) los que el campo vectorial
\[ f(x, y)=\left(4 x \sin (x-y), \frac{4 x+3 y}{2 x-y}\right) \nonumber \]
es continuo.
Ejercicio\(\PageIndex{6}\)
Para qué puntos en\(\mathbb{R}^n\) es el campo vectorial
\[ f(\mathbf{x})=\frac{\mathbf{x}}{\log (\|\mathbf{x}\|)} \nonumber \]
una función continua?
- Contestar
-
\(\left\{\mathbf{x}: \mathbf{x} \in \mathbb{R}^{n}, \mathbf{x} \neq \mathbf{0}\right\}\)