
6.1.E: Problems on Directional and Partial Derivatives


Exercise $$\PageIndex{1}$$

Complete all missing details in the proof of Theorems 1 to 3 and Corollaries 1 and 2.

Exercise $$\PageIndex{2}$$

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.

Exercise $$\PageIndex{3}$$

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.

Exercise $$\PageIndex{4}$$

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.)

Exercise $$\PageIndex{5}$$

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.)

Exercise $$\PageIndex{6}$$

$$\Rightarrow$$ 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.

Exercise $$\PageIndex{7}$$

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 $$k$$.]

Exercise $$\PageIndex{8}$$

State and prove Theorems 4 and 5 of Chapter 5, §1, for directional derivatives.