# Meaning of "Jacobian determinant"?

In his book on optimal control, George Leitmann uses a meaning of "Jacobian determinant" I don't understand (and I do understand the standard definition). Can anyone explain it? In particular, which function's Jacobian is being taken? (Also, why? I understand this why bit might be hard without more context.) The relevant excerpts from the book are, in order:

http://imgur.com/2vKMmkF,2WDQM91#0

http://imgur.com/2vKMmkF,2WDQM91#1

EDIT: I'm not requiring a firm answer like "This is definitely what it means." Even conjecture is fine.

• This might be useful: en.wikipedia.org/wiki/…
– Abel
May 31 '13 at 11:02
• Hi Abel, thanks for the link. I am aware of that definition -- but given that definition, can you tell me what Leitmann actually means? For example, after the word "namely", which function's Jacobian is that? The entries don't look like a normal Jacobian... May 31 '13 at 11:04

This so called Jacobian determinant is simply the Wronskian (up to a transposition). Why he has chosen to use that particular language, i'm not sure.

• Mmm! This is what I thought too when I first saw it. The question then becomes: How does this fit into his overall reasoning? After all, the result he wants to prove is exactly that the Wronskian $is$ zero (i.e. $g_{\alpha}$, $g_{\beta}$ are linearly independent.) May 31 '13 at 12:00
• Ah, Wronskian NON-zero implies linear independence. May 31 '13 at 12:23
• Sorry, my mistake. May 31 '13 at 12:45

In the determinant symbol,

$g_{\alpha}$ is the partial derivative of the function with respect to $\alpha$ and $g_{t\alpha}$ is the partial derivative with respect to $t$ of the function $g_{\alpha}$ and similarly for $\beta$. The determinant has the value

$$g_{\alpha}g_{t\beta} - g_{\beta}g_{t\alpha}$$.

So $g_{t\beta}$ for example, really means

$$\frac{\partial^{2}g}{{\partial t}{\partial \beta}}$$

• You have explained all the pieces of the notation acceptably. However, my question is $\textit{why}$ this is a Jacobian; or, more specifically, which function's Jacobian is this? May 31 '13 at 11:29
• You should probably modify your question to state clearly what you understand what you do not, otherwise people will answer what you already know! May 31 '13 at 11:33
• Sorry for not being clear, and thanks for your input so far. May 31 '13 at 11:36