Why is the trace of jacobian=the divergence?

[marellasunny]

What is the theoretical connection intuitively justifying that the trace of the jacobian=the divergence of a vector field? I know that this also equals the volume flow rate/original volume in the vector field but leaving that aside, what is the mathematical background behind establishing this equality?

 

[WannabeNewton]

Are you asking for a proof? The proof is trivial (it follows immediately from the form of the jacobian). Let $U\subseteq \mathbb{R}^{n}$ be open and let $F:U\rightarrow \mathbb{R}^{m}$ be differentiable at $a\in U$. Then the matrix representation of the total derivative of $F$ at $a$, in the standard basis $S$, is given by $(DF(a))_{S} = (\frac{\partial F^j}{\partial x^i}(a))$. We call this matrix representation the jacobian of such a map. Hence $Tr(DF(a))_{S} = \sum\frac{\partial F^{i}}{\partial x^{i}}(a)$. Thus if $X:U\subseteq \mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a vector field and $a\in U$, $Tr(DX)_{S}(a) = \frac{\partial X^{1}}{\partial x^1}(a) + \frac{\partial X^{2}}{\partial x^2}(a) + \frac{\partial X^{3}}{\partial x^3}(a) = (\nabla\cdot X)(a)$.

 

[lurflurf]

Well the Jacobian it the derivative of the function in the f:V->V senses. So any other derivative can be found from it, in particular we like invariants of the Jacobian including the trace and determinant as they are coordinate independent and as such represent the function without interference.

 

[marellasunny]

in particular we like invariants of the Jacobian including the trace and determinant as they are coordinate independent and as such represent the function without interference.

I think you meant "the trace and determinant are independent of the coordinate system".

Could you please explain the meaning of "represent the function without interference"?

 

[lurflurf]

coordinate independent and independent of the coordinate system mean the same thing, but coordinate independent is shorter and sounds better. By "represent the function without interference" I mean that sometimes one uses coordinate dependent, but when one does she must always consider what is caused by the function and what is caused by the coordinate system. A condition must be considered in terms of the coordinates. A simple example if we define a derivative (x^2f)' the condition f'=0 becomes ((x^2f)'-2xf)/x^2=0 when might cause confusion.