6.1.E: Problems on Directional and Partial Derivatives
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.