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Derivatives of scalar valued functions of second-order tensors

Let f (S) {\displaystyle f({\boldsymbol {S}})} be a real valued function of the second order tensor S {\displaystyle {\boldsymbol {S}}}. Then the derivative of f (S) {\displaystyle 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 second order tensor defined as

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

Properties: