- If $F \colon (N,[\omega_N]) \to (M, [\omega_M]) $ is an orientation-preserving diffeomorphism what can you say about $F^* \omega_M$? Hint. You get some condition on the determinant of a Jacobi matrix.
- Knowing the property of the determinant, use now the chain rule to ensure that the transition maps between charts in the atlas on $N$ that you obtain satisfy the requirements for this new atlas to be oriented.
Edit: The above is an attempt to stimulate some thinking on the problem. More specifically, one needs to have all the definitions handy in order to see that it is actually
much easier than it seems to be at the first sight. Apparently, this question has been
duplicated, so we now have some more information on the necessary background :-)
Now let me present a more elaborated answer.
This question is to solve Problem 20.3 from L.Tu "Introduction to Manifolds", p.209.
The key concept for this problem is when an atlas specifies the orientation on an oriented manifold. This notion is explained in the bottom of p. 207 of the cited textbook. I shall try to rephrase this in my own words.
Definition. An atlas $\{ (V_{\alpha}, \psi_{\alpha} ) \}_{\alpha \in A} $ specifies the orientation $[\omega_M]$ of an oriented manifold $(M, [\omega_M])$ if
$$
\psi_{\alpha}^* \epsilon^n \in [\omega_M |_{V_{\alpha}}]
$$
for each $\alpha \in A$, where $\epsilon^n$ is the standard Euclidean volume form, that is
$$
\epsilon^n := \mathrm{d} r^1 \wedge \dots \wedge \mathrm{d} r^n
$$
and $\omega_M |_{V_{\alpha}}$ is the restriction of a form $\omega_M$ onto $V_{\alpha}$.
(Functions $r^i$ here are the standard Euclidean coordinates on $\mathbb{R}^n$)
In the problem we are asked to show that the atlas $\{ (U_{\alpha}, \phi_{\alpha}) \}_{\alpha \in A}$ on manifold $N$, constructed so that $U_{\alpha} := F^{-1} V_{\alpha}$ and $\phi_{\alpha} := \psi_{\alpha} \circ F$ for each $\alpha \in A$, specifies the orientation $[F^* \omega_N]$ on $M$.
This brings us very close to the solution. I must stop here as this question is tagged as a homework, but really one needs just check that we have some everywhere positive functions involved.
In particular, there is no need to check Jacobians!
Best Answer
HINT:
If the Jacobian matrix has negative determinant then it is orientation reversing. If it has positive determinant then it is orientation preserving.
The Jacobian matrix, in this case, is the two-by-two matrix whose columns are $F_{r}$ and $F_{\theta}$. Can you find the partial derivatives, put them in a matrix and find its determinant?
You may find that the Jacobian is singular at some point.