27.1.1 Vector Identities
There are many interesting identities which relate the gradient, divergence and
Theorem 27.1.3 Assuming f,g are a C2 vector fields whenever necessary,
the following identities are valid.
- ∇⋅ = 0
- ∇×∇ϕ = 0
- ∇× =
−∇2f where ∇2f is a vector field whose ith component
- ∇⋅ =
- ∇× =
Proof: These are all easy to establish if you use the repeated index summation
convention and the reduction identities.
This establishes the first formula. The second formula is done similarly. Now consider the
This establishes the third identity.
Consider the fourth identity.
This proves the fourth identity.
Consider the fifth.
and this establishes the fifth identity. ■