Define a function z = f(x,y) by f(0,0) = 0 and otherwise?

[matthewturner]

please help me with this. At least some thing to start solving...

Define a function z = f(x,y) by f(0,0) = 0 and otherwise.

f(x,y) =(x^2 y) / (x^2+y^2 )

a. Show that in polar coordinates this function may be expressed (for r≠ 0) as z = r 〖cos〗^2 (θ)sin(θ)

b. Show that if θ is fixed then the graph is given by z = mr, a line of slope
m= 〖cos〗^2 (θ)sin(θ).
(Note that this says that the surface z = f(x,y) is what is called a ruled surface.)

c. Compute the directional derivatives of z in the θ direction. Does Df exist at the point (0,0)? Explain.

 

[tiny-tim]

welcome to pf!

hi matthew! welcome to pf!

(try using the X² button just above the Reply box )

show us what you've tried, and where you're stuck, and then we'll know how to help!

start with "a."

 

[DonAntonio]

please help me with this. At least some thing to start solving...

Define a function z = f(x,y) by f(0,0) = 0 and otherwise.

f(x,y) =(x^2 y) / (x^2+y^2 )

a. Show that in polar coordinates this function may be expressed (for r≠ 0) as z = r 〖cos〗^2 (θ)sin(θ)

b. Show that if θ is fixed then the graph is given by z = mr, a line of slope
m= 〖cos〗^2 (θ)sin(θ).
(Note that this says that the surface z = f(x,y) is what is called a ruled surface.)

c. Compute the directional derivatives of z in the θ direction. Does Df exist at the point (0,0)? Explain.

Start for noting that in polar coordinates, x=r\cos\theta\,\,,\,\,y=r\sin\theta , so substitute and get (a) at least . Now show some self effort.

DonAntonio

 

[matthewturner]

Thanks DonAntonio,

thanks for showing me the path. I almost completed the part a and b. have a confusion about the part c. here Df means the derivative of the f. Do you have any idea about that one?

Thanks again for helping me. I appreciate what you did. As you gave me before I just need a hint to proceed, not the whole proof. thanks again.