Calculus-Sec-13-3-Solution 13.3    Partial Derivatives                   by SGLee - HSKim, VLang, DYKim  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 . 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 . 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 .