6.1.E: Problems on Directional and Partial Derivatives
- Page ID
- 24086
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Complete all missing details in the proof of Theorems 1 to 3 and Corollaries 1 and 2.
Complete all details in Examples (a) and (b). Find \(D_{1} f(\vec{p})\) and \(D_{2} f(\vec{p})\) also for \(\vec{p} \neq 0.\) Do Example (b) in two ways: (i) use Note 3; (ii) use Definition 2 only.
In Examples (a) and (b) describe \(D_{\vec{u}} f : E^{2} \rightarrow E^{1}\). Compute it for \(\vec{u =(1,1)=\vec{p}.\)
In (b), show that \(f\) has no directional derivatives \(D_{\vec{u}} f(\vec{p})\) except if \(\vec{u} \| \vec{e}_{1}\) or \(\vec{u} \| \vec{e}_{2}.\) Give two proofs: (i) use Theorem 1; (ii) use definitions only.
Prove that if \(f : E^{n}\left(C^{n}\right) \rightarrow E\) has a zero partial derivative, \(D_{k} f=0,\) on a convex set \(A,\) then \(f(\vec{x})\) does not depend on \(x_{k},\) for \(\vec{x} \in A.\) (Use Theorems 1 and 2.)
Describe \(D_{1} f\) and \(D_{2} f\) on the various parts of \(E^{2},\) and discuss the relative continuity of \(f\) over lines through \(\overrightarrow{0},\) given that \(f(x, y)\) equals:
\[\begin{array}{ll}{\text { (i) } \frac{x y}{x^{2}+y^{2}};} & {\text { (ii) the integral part of } x+y;} \\ {\text { (iii) } \frac{x y}{|x|}+x \sin \frac{1}{y};} & {\text { (iv) } x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}};} \\ {\text { (v) } \sin (y \cos x);} & {\text { (vi) } x^{y}.}\end{array}\]
(Set \(f=0\) wherever the formula makes no sense.)
\(\Rightarrow\) Prove that if \(f : E^{\prime} \rightarrow E^{1}\) has a local maximum or minimum at \(\vec{p} \in E^{\prime},\) then \(D_{\vec{u}} f(\vec{p})=0\) for every vector \(\vec{u} \neq \overrightarrow{0}\) in \(E^{\prime}.\)
[Hint: Use Note 3, then Corollary 1 in Chapter 5, §2.
State and prove the Finite Increments Law (Theorem 1 of Chapter 5, §4) for directional derivatives.
[Hint: Imitate Theorem 2 using two auxiliary functions, \(h\) and \(k\).]
State and prove Theorems 4 and 5 of Chapter 5, §1, for directional derivatives.