I would like to prove that two finite-dimensional vector fields $\mathbf{A}=\mathbf{A}(\mathbf{x},t)$ and $\mathbf{B}=\mathbf{B}(\mathbf{x},t)$, with $\mathbf{x} \in \mathbb{R}^n$ and $t \in \mathbb{R}$, are the same vector field, i.e. $\mathbf{A}=\mathbf{B}$
Due to the nature of $\mathbf{A}$ and $\mathbf{B}$ expressions, might be very easy to show for example that $\nabla\cdot\mathbf{A}=\nabla\cdot\mathbf{B}$ or that $\nabla\times\mathbf{A}=\nabla\times\mathbf{B}$.
Are there some sufficient conditions involving nabla operator $\nabla$, or vector identities in general, to show that two given vector are the same vector (or they just differ from a constant)?
Remember that for a vector field $\mathbf V$: $$ \nabla \cdot(\nabla \times \mathbf V)=0 $$ and$$ \nabla \times(\nabla \mathbf V)=0 $$ so: $$ \nabla \cdot \mathbf A=\nabla \cdot(\mathbf A+\nabla \times \mathbf V) $$ and $$ \nabla \times \mathbf A=\nabla \times(\mathbf A+\nabla \mathbf V) $$