State whether each of these functions is symmetric, antisymmetric, or neither, with respect to the interchange of coordinates 1 and 2: (a) g(1)f(2)
b)(r1-r2)e^-br12 where r1 and r2 are vectors describing positions of two points and r12 is the distance between them |r1-r2|
To determine whether a function is symmetric, antisymmetric, or neither with respect to the interchange of coordinates 1 and 2, we can replace coordinates 1 and 2 and see if the function remains unchanged.
(a) g(1)f(2):
To evaluate whether this function is symmetric or antisymmetric, we replace coordinates 1 and 2:
g(2)f(1)
If g(2)f(1) equals g(1)f(2), then the function is symmetric.
If g(2)f(1) equals -g(1)f(2), then the function is antisymmetric.
If g(2)f(1) is neither equal to nor the negation of g(1)f(2), then the function is neither symmetric nor antisymmetric.
(b) (r1 - r2)e^(-br12):
To determine the symmetry of this function, let's replace coordinates 1 and 2 by switching their places:
(r2 - r1)e^(-br21)
If (r2 - r1)e^(-br21) equals (r1 - r2)e^(-br12), then the function is symmetric.
If (r2 - r1)e^(-br21) equals -(r1 - r2)e^(-br12), then the function is antisymmetric.
If (r2 - r1)e^(-br21) is neither equal to nor the negation of (r1 - r2)e^(-br12), then the function is neither symmetric nor antisymmetric.
Remember, in both cases, r1 and r2 are vectors describing the positions of two points, and r12 is the distance between them, |r1 - r2|.
By evaluating the expressions g(2)f(1) and (r2 - r1)e^(-br21), you can determine the symmetry of the given functions.