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What is Jacobian Matrix?

Is there any physical intuition for the Jacobian?

I understand that it is the matrix of partial derivatives and how to construct it. What I want to know is

  • what's the use of it? Application wise
  • is there a nice intuitive explanation for it? I mean regarding its significance and otherwise

marked as duplicate by Rahul, Sasha, user31373, Dylan Moreland, Zev Chonoles Jul 6 '12 at 1:07

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  • 1
    $\begingroup$ There is a geometric intuition: this is how a small volume changes after a coordinate change. $\endgroup$ – Artem Jul 5 '12 at 19:29
  • $\begingroup$ What about it's application? Where is it used? $\endgroup$ – Inquest Jul 5 '12 at 19:34
  • $\begingroup$ Wikipedia offers useful information. $\endgroup$ – Raymond Manzoni Jul 5 '12 at 19:34

The Jacobian is the matrix that represents the linear transformation that takes a small change in the input of a function to the corresponding small change in output:

For $f:\mathbb R^n\to\mathbb R^m$, a fixed $x \in \mathbb R^n$, we have $$f(x+h) = f(x) + J(x)h + o(|h|) \qquad\text{for }h\in\mathbb R^m, h\to 0$$ provided that the various partial derivatives exist and behave sufficiently nicely.

In this way the Jacobian is the direct analogue of the derivative in ordinary real analysis.


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