Polar coordinates

[Imum Coeli]

Question:
a) Find a function f with domain R^2 is such that, when expressed in polar coordinates (r,theta), it does not depend on theta. Carefully explain what the level curves look like.

b) What is the (implied) domain of g(x,y) = 1/sin(x*y)?

Notes:
Not sure if this is right but...

a)
Let f(x,y) = x^2 + y^2
Then changing to polar coordinates gives
f(x,y) = (r*cos(theta))^2 + (r*sin(theta))^2
f(x,y) = r^2*(cos(theta)^2 + sin(theta)^2)
f(x,y) = r^2
therefore f(x,y) does not depend on theta.

b)
Simply need sin(x*y) ~= 0
=> x,y belong to R\{0}

Seems a bit too simple...

 

[HallsofIvy]

Question:
a) Find a function f with domain R^2 is such that, when expressed in polar coordinates (r,theta), it does not depend on theta. Carefully explain what the level curves look like.

b) What is the (implied) domain of g(x,y) = 1/sin(x*y)?

Notes:
Not sure if this is right but...

a)
Let f(x,y) = x^2 + y^2
Then changing to polar coordinates gives
f(x,y) = (r*cos(theta))^2 + (r*sin(theta))^2
f(x,y) = r^2*(cos(theta)^2 + sin(theta)^2)
f(x,y) = r^2
therefore f(x,y) does not depend on theta.i

Yes, that is a correct answer.

b)
Simply need sin(x*y) ~= 0
=> x,y belong to R\{0}

Seems a bit too simple...

It is. Suppose \(\displaystyle x= \pi\), y= 1? There are many other values of x and y that make sin(xy)= 0.

 

[Imum Coeli]

Ah of course! Thanks.

So... Does that mean the domain is x ~= 0 and y ~= k*pi/x for k belonging to the integers?
Or do I have to pick a smaller domain on which it is continuous? (My notes say pick the largest subset of R^2 which is possible.)

 

[HallsofIvy]

sin(xy) is symmetric in x and y. If "x not equal to 0 and y not equal to kpi/x" is correct then so is "y not equal to 0 and x not equal kpi/y". I would be inclined to just write "xy not an integer multiple of pi".