1

I am following a derivation in a Calculus of Variation problem. After introducing a one-parameter family of one-to-one mappings from $R^{2}$ to itself, $$z({x},\epsilon)$$, $x = (x_1,x_2)$, such that $$z({x},0) = x$$. The mapping is used for a change of variable within an integral, hence the issue of computing the variation of the jacobian's determinant arises. The author says that the obvious identity $$\delta \, det z_{a,b} = \delta z_{a,a}$$

Well as it often happens, it is not entirely obvious to some...Any help would be so appreciated, thanks

An aedonist
  • 2,568
  • 2
    What is $z_{a,a}$ exactly? Please add a reference to the source of the claim. The author could be using some structure of $z$ that you did not include in the post. –  Jun 30 '14 at 00:07
  • The source is page 39 of the book "Introduction to Micromechanics" by K. Le.Repeated indices summation is used, so the way I read it $z_{a,a} = z_{1,1}+z_{2,2}$, the comma indicated partial derivation. – An aedonist Jun 30 '14 at 11:53
  • $z_{ab}$ is obviously a matrix--we need to know what that matrix is. – Jared Jul 01 '14 at 04:17

1 Answers1

1

Hints:

  1. Consider a differentiable function $f: \mathbb{R}^n \times \mathbb{R}\to \mathbb{R}^n$ [let us write $z=f(x,\epsilon)$] with the property that $f(\cdot , 0)={\rm id}_{\mathbb{R}^n}$ is the identity map.

  2. Define Jacobi matrix $A:=\frac{\partial f}{\partial x}$. Then $\left. A \right|_{\epsilon=0}={\bf 1}_{n\times n}$ is the identity matrix.

  3. One can argue in several ways that the sought-for identity must hold, e.g. $$ \left.\frac{\partial}{\partial\epsilon} \det(A) \right|_{\epsilon=0} ~=~\left. \det(A)^{-1} \frac{\partial}{\partial\epsilon} \det(A) \right|_{\epsilon=0} ~=~\left.\frac{\partial}{\partial\epsilon}\ln\det(A) \right|_{\epsilon=0}$$ $$~=~\left.\frac{\partial}{\partial\epsilon}{\rm tr}\ln(A) \right|_{\epsilon=0} ~=~\left.{\rm tr}(A^{-1}\frac{\partial}{\partial\epsilon}A) \right|_{\epsilon=0} ~=~\left.\frac{\partial}{\partial\epsilon}{\rm tr}(A) \right|_{\epsilon=0}. $$

Qmechanic
  • 12,298
  • Qmechanic, thank you very much as usual. I am in the process of deciphering your derivation, as I am still unfamiliar to certain tensorial operations. Let me then try to learn more by checking why I am unable to derive the same by using the definition. Let introduce a variation in $z$ by setting $\hat{z} = z + \epsilon \bar{z}$. – An aedonist Jul 01 '14 at 14:17
  • Qmechanic, thank you very much as usual. I am in the process of deciphering your derivation, as I am still unfamiliar to certain tensorial operations. Let me then try to learn more by checking why I am unable to derive the same by using the definition. stay in $R^2$. Let us introduce a variation in $z$ by setting $\hat{z} = z + \epsilon \bar{z}$. Then the variation of the determinant equals $ \delta \operatorname{det} z_{a,b} = \lim_{\epsilon \to 0} \frac{\operatorname{det} \hat{z} - \operatorname{det}{z}}{\epsilon} =$ (To be continued) – An aedonist Jul 01 '14 at 14:30
  • Hi @Buco. You can only edit a comment for 5 minutes but you can delete it anytime. So the trick (for a LaTeX-heavy comment) is to store the comment in an external editor, copy&paste into a new comment, edit for 5 min, delete, open a new comment, edit for 5 min, delete, open a new comment, etc, until you are satisfied with the format of the comment. Alternatively, you can try flag a moderator, but I doubt he would be too enthusiastic about having to clean up other peoples comments. – Qmechanic Jul 02 '14 at 20:20
  • @Buco: Hover the mouse over the comment you want to delete. An "x" should appear at the end of the comment. Click on the "x". Bam! – Qmechanic Jul 02 '14 at 22:29
  • $$\lim_{\epsilon \to 0} \frac{\left(z_{1,1}+\epsilon \bar{z}{1,1}\right)\left(z{2,2}+\epsilon \bar{z}{2,2}\right)-\left(z{1,2}+\epsilon \bar{z}{1,2}\right)\left(z{2,1}+\epsilon \bar{z}{2,1}\right)-\operatorname{det}z{a,b}}{\epsilon}$$ Simplifying, one gets $$\delta z_{a,b} =\bar{z} {1,1} z{ 2,2} +\bar{z}{ 2,2} z{ 1,1} −z_{ 1,2} \bar{z} _ {2,1} -z_{2,1} \bar{z}_{ 1,2},$$ not the trace. Where do things go wrong? – An aedonist Jul 02 '14 at 22:50
  • @Buco, I have fixed the formatting on your comment. – Alexander Gruber Jul 10 '14 at 03:14