Directional derivative
Derivatives of tensor valued functions of second-order tensors

Let F (S) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} be a second order tensor valued function of the second order tensor S {\displaystyle {\boldsymbol {S}}}. Then the derivative of F (S) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} with respect to S {\displaystyle {\boldsymbol {S}}} (or at S {\displaystyle {\boldsymbol {S}}}) in the direction T {\displaystyle {\boldsymbol {T}}} is the fourth order tensor defined as

for all second order tensors T {\displaystyle {\boldsymbol {T}}}.

Properties:
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