Calculus-Sec-13-3-Solution


   13.3    Partial Derivatives                   by SGLee - HSKim, VLang, DYKim

                                                                     http://youtu.be/rSYLp1mSMXY

 

 1. Find partial derivatives with respect to  and   for the function .

 

      .




2. Find partial derivatives with respect to  and  for the function .

 

      .

3. Find all second order partial derivatives of the function .

 

 

      

     

     .

4. Find all second order partial derivatives of the function .

 

 

     

     

     .

 5. Take an example of the  solution for the .

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

 

 




6-8. Laplace’s equation A classical equation of mathematics is Laplace’s equation,

       which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials,

       and the steady-state distribution of heat in a conducting medium.

       In two dimensions, Laplace’s equation is

                 

       Show that the following functions are harmonic; that is, they satisfy Laplace’s equation.

 

 6. 

 




7.  for any real number .

 




8. 

 




9. The volume of a right circular cone of radius  and height  is .

    Show that if the height remains constant while the radius changes the volume satisfies .

 




10. Show that the function defined by  is not continuous at  but its first order partial derivatives exist at .

 

참고 ( http://calculus.subwiki.org/wiki/Existence_of_partial_derivatives_not_implies_differentiable)

By the definition of partial derivatives, we have

 , 

 

,

However, f(x,y) is not continuos at (0,0), that is,

Let x->0, and y->0 along the line y=mx, so the limit of f(x,y) does not exist.

For m=0, f(x,y)->0 and for m=1, f(x,y) -> 1/5




Show that the function defined by  is continuous at  but its first order partial derivatives do not exit at .

 

 

Since the definition of partial derivatives, we have

 ,  (undefined)

 

,

 

However, f(x,y) is continuos at (0,0), that is,

 




12. Let us consider the function . Show that  at .

 

 

 , 

 

, (undefined)

 

So,  at .

 




3. Let us consider the function . Show that  at .

 

 

 , 

 

, (undefined)

 




14. Let . Show that .

                                                      Back to Part II