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For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$?

- $\overrightarrow{A}$ is solenoidal if $\triangledown \cdot \overrightarrow{A}=0.$
- $\triangledown \times \overrightarrow{A}$ is another vector field.
- $\overrightarrow{A}$ is irrotational if $\triangledown ^{2}\overrightarrow{A}=0$.
- $\triangledown\times \left ( \triangledown \times \overrightarrow{A} \right)=\triangledown \left ( \triangledown \cdot \overrightarrow{A} \right )-\triangledown ^{2}\overrightarrow{A}$