Question

What's the 'symmetric' counterpart to an integral?

Answer 1

*

Answer 2

If we have some matrix A we can look at its members \[\Large a_{mn}\] and if A is symmetric, then \[\Large a_{mn}=a_{nm}\] Similarly if it's skew symmetric then \[\Large a_{mn}=-a_{nm}\] So if w look at an integral, we have: \[\Large \int\limits_m^n f(x)dx= g(n,m)\] and we can similarly see that \[\Large g(n,m)=-g(m,n)\] But where's the symmetric part?

Answer 3

I immediately think of the conjugate property for inner products defined for functions, like
\[\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}\,dx=\overline{\langle g,f\rangle}\]
but I don't know if that's relevant...

Answer 4

why Im bad in math and physics?