How are even and odd functions defined in n-dimensions?

[PeteyCoco]

How are even and odd functions defined in n-dimensions?

In my homework, we had to integrate f(x,y,z)= (x^2)z + (y^2)z + z^3 over a sphere centered at the origin. My answer came out to be 0 and I made the guess that it might be because f(x,y,z) was an odd function. Now, I don't know if this is true. Does this assumption hold:

f(x) is odd if f(-x) = -f(x)
f(x,y,z) is odd if f(-x,-y,-z) = -f(x,y,z) ?

I'm thinking this might just mean it's odd across a plane.I just handed in my assignment and will have to wait 2 weeks before I know if what I did was right or wrong, so I'm asking here to speed things up!

 

[SammyS]

How are even and odd functions defined in n-dimensions?

In my homework, we had to integrate f(x,y,z)= (x^2)z + (y^2)z + z^3 over a sphere centered at the origin. My answer came out to be 0 and I made the guess that it might be because f(x,y,z) was an odd function. Now, I don't know if this is true. Does this assumption hold:

f(x) is odd if f(-x) = -f(x)
f(x,y,z) is odd if f(-x,-y,-z) = -f(x,y,z) ?

I'm thinking this might just mean it's odd across a plane.I just handed in my assignment and will have to wait 2 weeks before I know if what I did was right or wrong, so I'm asking here to speed things up!

I'm not sure about whether we refer to functions of more than one variable as even or odd in an overall sense, although I doubt that we do.

However, it's common to refer to a function of more than one variable as being even or odd with respect to an individual variable.

Factor z out of your function: f(x, y, z) = z(x² + y² + z²) .

 

[vela]

f(x,y,z) is odd if f(-x,-y,-z) = -f(x,y,z) ?

The parity transformation takes (x,y,z) to (-x,-y,-z). It's common to say f has even or odd parity if f(-x,-y,-z) is equal to, respectively, f(x,y,z) or -f(x,y,z). But as SammyS noted, when you have more than one variable, it's ambiguous if all you say is "f is odd." You should specify under what type of transformation is the function odd.

 

[haruspex]

Since the region of integration is symmetric in all three coordinates, the integral will be zero if the function is odd wrt any of the three.