I have question about (b) and (c) (b) ) What is the domain of the function f abo
ID: 2889216 • Letter: I
Question
I have question about (b) and (c)
(b) ) What is the domain of the function f above? That is what is the largest subset of R3 on which f is defined.
the answer is everything except z-axis D= {(x,y,z) | (x,y) not = (0,0)}
I understand x and y shouldn't be 0 but why z-axis ???
(c) Is the domain from above open? Is it connected? Is it simply connected? You must give an explanation behind your answer for each questions.
open = no boundary
connected = I understand that it is available to draw a curve, but why z-axis should be avoided???
Not simply connected, but why????
-2y F =Explanation / Answer
(b) Domain D is given by,
D = [ (x,y,z) | (x,y) not = (0,0) and z can be any real numbaer (for all z ).
Reason for z-axis :
Because function F is independent of x,y and z ie constant or we can say that the z component of function F will always be equal to 1.
For Example : Let us assume any point say (1,2,3) [ except (x,y) = (0,0)]
Now Function F can be writtent as,
F = < (-2 / 25), (-4 / 25), 1 >
Here, we can see that we have chosen point (1,2,3) { z=3} but the z component of function F is 1.
Let us take another point (2,5,0)
when we calculate the z component of function F, again it will be 1 as it is independent of x, y and z.
Hence there is z axis in the domain.
(c) Domain is open as it does not contains the Z-axis. z- axis is a boundary points.
Note : Working on part (c) connected and simply connected domains and upload the explanation soon.