There are many vector differential identities linking the three vector differential operators (gradient, divergence and curl) to one another. They are extremely useful in applied mathematics and theoretical physics.

Let f be any scalar function of position, and let **A** and **B** be vector functions of position. Then the following statements are true:

div(curl(**A**)) = 0
curl(grad(**A**)) = 0
div(f**A**) = (**A**.grad)f + fdiv(**A**)
curl(f**A**) = grad(f)^**A** + fcurl(**A**)
div(**A**^**B**) = **B**.(curl(**A**)) - **A**.(curl(**B**))
curl(**A**^**B**) = (**B**.grad)**A** - **B**(div(**A**)) - (**A**.grad)**B** + **A**(div(**B**))
grad(**A**.**B**) = (**B**.grad)**A** + **B**^(curl(**A**)) + (**A**.grad)**B** + **A**^(curl(**B**))
div(grad(**A**)) = grad(div(**A**)) - curl(curl(**A**)).

All these results can be proved using

summation convention. For example, consider the fifth expression above.

div(**A**^**B**) = d_{i}(ε_{ijk}A_{j}B_{k})
= ε_{ijk}d_{i}(A_{j}B_{k})
= ε_{ijk}(B_{k}d_{i}A_{j} + A_{j}d_{i}B_{k})
= B_{k}(ε_{kij}d_{i}A_{j}) - A_{j}(ε_{jik}d_{i}B_{k})
= **B**.(curl(**A**)) - **A**.(curl(**B**)).