17.1.1 Vector Identities
There are many interesting identities which relate the gradient, divergence and curl.
Theorem 17.1.3 Assuming f,g are a C2 vector fields whenever necessary, the following identities
- ∇⋅ = 0
- ∇×∇ϕ = 0
- ∇× =
−∇2f where ∇2f is a vector field whose ith component is ∇2fi.
- ∇⋅ =
- ∇× =
Proof: These are all easy to establish if you use the repeated index summation convention and the
This establishes the first formula. The second formula is done similarly. Now consider the third.
This establishes the third identity.
Consider the fourth identity.
This proves the fourth identity.
Consider the fifth.
and this establishes the fifth identity. ■