##### Second Derivatives
c.m.dc.d2

###### Divergence of gradient

\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the Laplacian of the scalar field \(T\). The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components given by the Laplacian of the corresponding vector elements.

###### Curl of a gradient

This always vanishes.

###### Gradient of the divergence

\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.

###### Divergence of a curl

This always vanishes.

###### Curl of curl

\[ {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v} \tag{curlcurl}\label{curlcurl} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42