04 - tensor calculus - tensor analysis 04 - tensor calculus 1 tensor analysis - frechet derivative • consider smooth differentiable scalar field with scalar argument vector argument tensor argument • frechet derivative (tensor notation) scalar argument vector argument tensor argument tensor calculus 2 tensor analysis - gateaux derivative • consider smooth differentiable scalar field with scalar argument vector argument tensor argument • gateaux derivative,i.e.,frechet wrt direction (tensor notation) scalar argument vector argument tensor argument tensor calculus 3 tensor analysis - gradient • consider scalar- and vector field in domain • gradient of scalar- and vector field renders vector- and 2nd order tensor field tensor calculus 4 tensor analysis - divergence • consider vector- and 2nd order tensor field in domain • divergence of vector- and 2nd order tensor field renders scalar- and vector field tensor calculus 5 tensor analysis - laplace operator • consider scalar- and vector field in domain • laplace operator acting on scalar- and vector field renders scalar- and vector field tensor calculus 6 tensor analysis - transformation formulae • consider scalar,vector and 2nd order tensor field on • useful transformation formulae (tensor notation) tensor calculus 7 tensor analysis - transformation formulae • consider scalar,vector and 2nd order tensor field on • useful transformation formulae (index notation) tensor calculus 8 tensor analysis - integral theorems • consider scalar,vector and 2nd order tensor field on • integral theorems (tensor notation) green gauss gauss tensor calculus 9 tensor analysis - integral theorems • consider scalar,vector and 2nd order tensor field on • integral theorems (tensor notation) green gauss gauss tensor calculus 10 voigt / matrix vector notation • strain tensors as vectors in voigt notation • stress tensors as vectors in voigt notation • why are strain & stress different? check energy expression! tensor calculus 11 voigt / matrix vector notation • fourth order material operators as matrix in voigt notation • why are strain & stress different? check these expressions! tensor calculus 12.