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Let $X$ and $Y$ be locally compact Hausdorff spaces. A linear
map $T$ from $C_0(X)$ into $C_0(Y)$ is said to be a
disjointness preserving (or separating) if $f\cdot g=0$ implies
$Tf\cdot Tg=0$. We extend a result of K. Jarosz to describe
the general form of all such maps. In particular, if $T$ is
bounded then $T$ is essentially of the Banach-Stone form $Tf=
h\cdot (f\circ\varphi)$. By the Banach-Stone theorem, every
bijective linear isometry $T$ from $C_0(X)$ onto $C_0(Y)$ is a
disjointness preserving map. We prove in this paper, every
isometry $T$ from $C_0(X)$ into $C_0(Y)$ induces a linear
disjointness preserving isometry $T_1$ from $C_0(X)$ into $C_0(
Y_0)$ for some locally compact subset $Y_0$ of $Y$ such that
$T_1$ have the Banach-Stone form $T_1 f=Tf_{|_{Y_0}}=h\cdot(
f\circ \varphi)$ where $\varphi$ is a quotient map from $Y_0$
onto $X$. Let $E$ and $F$ be Banach spaces and $F$ be strictly
convex. We extend a result of M. Jerison to ensure that every
injective linear isometry from $C_0(X,E)$ into $C_0(Y,F)$
induces a disjointness preserving mapping $T_1$ from $C_0(X,E)$
into $C_b(Y_0,F)$ for some non-empty subset $Y_0$ of $Y$ such
that $T_1 f=Tf_{|_{Y_0}}=h\cdot(f\circ\varphi)$ where $\varphi$
is a continuous map from $Y_0$ onto $X$.
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