- #1
Marcus95
- 50
- 2
Homework Statement
Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds:
## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ##
Homework Equations
## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##
The Attempt at a Solution
My approach is the following:
we know that ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##
Applying the time derivative to this:
## \frac{d}{dt} det(A) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} (A_{il}'A_{jm}A_{kn}+ A_{il}A_{jm}'A_{kn}+ A_{il}A_{jm}A_{kn}') ##
## = det(A) (\frac{1}{A_{il}} A_{il}' +\frac{1}{A_{jm}} A_{jm}' + \frac{1}{A_{kn}} A_{kn}') ##
This is starting to look somewhat like the expression we look for, but from here on I am stuck. Any ideas on how to continue? Many Thanks!