Calculate the Poisson brackets {$f, g$} for
(a) f = q, g = $p^{3}$ + p - 1
(b) f = sin q , g = cos p
I am trying to do this via the definition of Poisson brackets (not sure who to write this in Latex)
I got 0 for both but but both answers are incorrect.
Since I see variables $(p,q)$ I suppose you are dealing with a 2-dimensional phase space with standard symplectic structure $\Omega=dq\wedge dp$. So you have $$ \{f,g\}=\Omega(X_f,X_g) $$ and this could be a way to compute the bracket, where $X_f,X_g$ are the Hamiltonian vector fields of the two functions.
Otherwise you can just use the coordinate based definition $$ \{f,g\} = (\partial_q f)(\partial_pg)-(\partial_pf)(\partial_qg). $$
So in the first exercise you have $$\{f,g\}=(1)(3p^2+1)=3p^2+1, $$ while in the second one you have $$\{f,g\}=(\cos{q})(-sin{p})=-\cos{(q)}\sin{(p)}. $$
Just to complete my answer, starting from the characterization $\{f,g\}=\Omega(X_f,X_g)$ you can think about the Poisson bracket even in geometrical terms and hence complete the computations in an alternative way: $$\Omega(X_f,X_g)=i_{X_f}X_g=df(X_g)=\mathcal{L}_{X_g}f$$ which is the Lie derivative of $f$ along $X_g$.
Let's do again the computations for the first example with this approach, just to check they are alternative procedures:
$$ X_g = \begin{bmatrix} 3p^2+1 \\ 0 \end{bmatrix} $$ $$ \nabla f = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $$ $$\{f,g\}=\mathcal{L}_{X_g}f=X_g\cdot \nabla f = 3p^2+1, $$ which confirms the previous (and easier) computation.
