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I am a first year undergraduate and I always see the Jacobian crop up in some many places, e.g., integration, solving systems of equations, analysis and so many more places. I was wondering, what makes it so useful?

Rodrigo de Azevedo
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PhysicsMathsLove
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    You have seen it in many places but you wonder why it is useful? It seems like you can solve your question yourself. –  May 17 '17 at 13:52
  • Well yes, I know what it does. But I was looking for someone to provide some reason as to what inherent property it has that allows it to be so useful. – PhysicsMathsLove May 17 '17 at 14:13

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Personally, I think that's the same as asking why the derivative is useful, which I'd assume you'd have an easier time intuiting.

In some sense, the Jacobian is the derivative of a multivariate vector function. I.e. if $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$, what is the derivative of $f$? The Jacobian encodes the derivatives of all components of $f$ with respect to all variables, and thus serves as its derivative. Notice that we recover the classic derivative when $n=m=1$ and the gradient when $n=1$. Dynamical systems, for example, can be treated as such a function, and linearized that way, using the property of the derivative (and Jacobian) as providing the best local linear approximation to a function.

The fact that it is structured as an array is also useful, as it lets it be used very naturally with tensors or matrices and vectors, so that useful equations and identities in $\mathbb{R}$ generalize in notationally pleasant ways to higher dimensions (and even to manifolds) via the Jacobian.

user3658307
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