12.2 Partial Derivatives and Directional Derivatives

1. Show that  .

(Sol)

Let , so .

Thus

   

2. If , find .

 http://matrix.skku.ac.kr/cal-lab/cal-12-2-2.html

(Sol)

   (x, y, z) |--> (-x/(x^2 + y^2 + z^2)^(3/2), -y/(x^2 + y^2 +

   z^2)^(3/2), -z/(x^2 + y^2 + z^2)^(3/2))

Therefore, .


3. If , find .


4. State and chat on the mean value Theorem for multi-variable functions.


5. Let with .

(a) Show that and exist everywhere.

(b) Are and continuous at the origin? Justify your answer.

  http://matrix.skku.ac.kr/cal-lab/cal-12-2-5.html

(Sol)

(a) If , .

  Else

-(x^2*y - y^3)/(x^4 + 2*x^2*y^2 + y^4)

(x^3 - x*y^2)/(x^4 + 2*x^2*y^2 + y^4)


Thus and exist everywhere.

(b)

0

Infinity


 is not continuous at the origin.

Infinity

0


 is not continuous at the origin.


6.(a) Take an example of the  solution for the .

   (b) In (a), Can you find the examples other than polynomial?

(Sol)

(a) http://matrix.skku.ac.kr/cal-lab/cal-12-2-6.html

 

([x == -y], [1])


(b)


7. Can you find and prove an analogue for Rolle's Theorem, for functions of two real variables?

(Sol)

Rolle's Theorem for a variable: If a real-valued function is continuous on a closed interval [a, b], differentiable on the open

interval (a, b), and , then there exists a c in the open interval (a, b) such that .