Continuity and Polar Coordinates

[schmidt7100]

Homework Statement

Show that the function f(x,y)= xy/sqrt(x^2+y^2) is continuous at the origin using polar coordinates. f(x,y)=0 if (x,y)=(0,0)

Homework Equations

r=sqrt(x^2+y^2)
x=rcos(theta)
y=rsin(theta)

The Attempt at a Solution

So, converting this equation to polar coordinates, I get rsin(theta)cos(theta). However, after this I'm stumped as to how I prove that this is continuous at the origin.

 

[rock.freak667]

At the origin in polar coordinates (r,θ), shouldn't r=0 and θ=0 ?

 

[HallsofIvy]

Not exactly- at the origin r= 0 and \theta is undefined.

However, Schmidt7100, the point is that r alone measures the distance of a point from the origin- \theta is irrelevant. That means that, if \lim_{r\to 0}f(r,\theta) does not depend on \theta, then that is the limit of f(r, \theta) as (r, \theta) goes to the origin.

Note that is NOT the case for Cartesian coordinates. Since the distance from (x, y) to (0, 0) depends upon both x and y. I am sure you have seen examples of functions in (x, y) that give different limits as you go to (0, 0) along different paths.

 

[hunt_mat]

You don't have to use polar co-ordinates, let
<br /> x_{n}=\frac{1}{n},\quad y_{n}=\frac{1}{n}<br />
Insert these into your equation for f and let n\rightarrow\infty, compute the limit and if it's the same as f(0,0) you've proved continuity.