Select all of the equations below in which $v$ is directly proportional to $w$.
$$v=\frac{3}{w}v=w+3v=\frac{w}{3}$$
$$v=3wv=w^3$$

Step 1 ：Analyze each equation to determine if it can be written in the form $v=kw$, where $k$ is a constant.

Step 2 ：$v=\frac{3}{w}$ cannot be written in the form $v=kw$, so it does not represent direct proportionality.

Step 3 ：$v=w+3$ cannot be written in the form $v=kw$, so it does not represent direct proportionality.

Step 4 ：$v=\frac{w}{3}$ can be written as $v=\frac{1}{3}w$, which is in the form $v=kw$. Therefore, it represents direct proportionality.

Step 5 ：$v=3w$ is already in the form $v=kw$, so it represents direct proportionality.

Step 6 ：$v=w^3$ cannot be written in the form $v=kw$, so it does not represent direct proportionality.

Step 7 ：$\boxed{{\displaystyle v=\frac{w}{3},v=3w}}$ are the equations in which $v$ is directly proportional to $w$.