Answer
$T_{rr}=4e^{-2r}cosθ$
$T_{θθ}=-e^{-2r}cosθ$
$T_{rθ}=T_{θr}=2e^{-2r}sinθ$
Work Step by Step
Take the first partial derivatives of the given function. When taking partial derivative with respect to r, treat θ as a constant, and vice versa: $T_{r}=-2e^{-2r}cosθ$
$T_{θ}=-e^{-2r}sinθ$
Then take the derivative of the first order partial derivatives to find second partial derivatives:
$T_{rr}=4e^{-2r}cosθ$
$T_{θθ}=-e^{-2r}cosθ$
Second partial derivatives of first order partial derivative of r with respect to θ and θ with respect to r are the same:
$T_{rθ}=T_{θr}=2e^{-2r}sinθ$
