Differential-Form Version of (div grad) V = curl curl V + grad div V
- From: are <Proponent@xxxxxxx>
- Date: Mon, 10 Dec 2007 03:11:49 -0800 (PST)
Given a vector field V, one has the identity
[1] (div grad) V = curl curl V + grad div V ,
where (div grad) denotes the Laplacian operator.
What is the equivalent of this identity in the language of
differential forms? Taking V now as a 1-form, it seems to me that the
right-hand side of the identity [1] might be expressed as
[2] d*dV + d*d*V .
On the other hand, if V is regarded as a 2-form, then we might write
[3] d*d*V + d*dV .
But what can one do with the left-hand side of equation [1]?
.
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