is a smooth orientable even-dimensional submanifold of $M$ and $$d_wf\big(T_w\big(f^{-1}(Y)\big)\big)= d_w(TM)\cap T_{f(w)}Y$$ for every $w\!\in\!f^{-1}(Y)$.

If $f\!:M\!\lra\!X$ intersects $Y\!\subset\!X$ transversally and $M$, $X$, and $Y$ are orientable of even total dimension, then $f$ intersects $Y$ properly. However, a proper intersection need not be transverse. For example, any two real lines in $\R^n$ intersect properly, but not transversally if $n\!\ge\!3$. Two curves that are tangent to each other do not intersect properly.\
If $f\!:M\!\lra\!X$ intersects $Y\!\subset\!X$ properly and $NY\!\lra\!Y$ is the normal bundle of $Y$ in $X$, the homomorphisms $$d_w^{NY}f\!: T_wM\lra N_{f(w)}Y, \quad v\lra d_wf(v)+T_{f(w)}Y,
\qquad w\!\in\!f^{-1}(Y),$$ have constant rank; the kernel of $d_w^{NY}f$ is $T_w(f^{-1}(Y))$. If $M$, $X$, and $Y$ are oriented, an orientation on $f^{-1}(Y)$ then induces an orientation on the vector bundle $$N^fY\equiv f^*NY\big/(\Im\, d^{NY}f)\lra f^{-1}(Y).$$ Note that N\^fY=(X-M)-(Y-f\^[-1]{}(Y)).\
Let $Y$ be a compact symplectic submanifold of $X$ and $$\io_{Y*}\!: H_*(Y;\Z)\lra H_*(X;\Z)$$ the homomorphism induced by the inclusion $\io_Y\!:Y\!\lra\!X$. If $\be_Y\!\in\!H_2(Y;\Z)$ and $J$ is an $\om$-tame almost complex structure on $X$ which preserves $TY\!\subset\!TX|_Y$, then $\io_Y$ induces an embedding $$\ov\M_{g,S}(Y,\be_Y;J)\hookrightarrow \ov\M_{g,S}(X,\io_{Y*}\be_Y;J).$$ If $f_j\!:M_j\!\lra\!X$, $j\!\in\!S$, are smooth maps as above, let $$\ov\M_{g,\f}(Y,\be_Y;J)=\big\{\big([u],(w_j)_{j\in S}\big)
\in \ov\M_{g,\f}(X,\io_{Y*}\be_Y;J)\!:\, [u]\!\in\!\ov\M_{g,S}(Y,\be_Y;J)\big\}.$$ If in addition $u\!:\Si_u\!\lra\!Y$ is a $J$-holomorphic map from a nodal Riemann surface 