In the world of linear approximations of multiple parameters and multiple outputs, the Jacobian is a matrix that tells you: **if I twist this knob, how does that part of the output change?**

(The Jacobian is defined at a point. If the space not flat, but instead only approximated by flat things that are joined together, then you would stitch together different Jacobians as you stitch together different flats.)

Pretend that **a** through **z** are parameters, or knobs you can twist. Let’s not say whether you have control over them (endogenous variables) or whether the environment / your customers / your competitors / nature / external factors have control over them (exogenous parameters).

And pretend that **F¹** through **Fⁿ** are the separate kinds of output. You can think in terms of a real number or something else, but as far as I know the outputs cannot be linked in a lattice or anything other than a matrix rectangle.

In other words this matrix is just an organised list of “how parameter **c** affects output **F⁹**”.

Notan bene — the Jacobian is just a linear approximation. It doesn’t carry any of the info about mutual influence, connections between variables, curvature, wiggle, womp, kurtosis, cyclicity, or even interaction effects.

A Jacobian tensor would tell you how twisting knob **a** knocks on through parameters **h**, **l**, and **p**. Still linear but you could work out the outcome better in a difficult system — or figure out what happens if you twist two knobs at once.

In maths jargon: the Jacobian is a matrix filled with partial derivatives.