Gradient Explained
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Gradient Explained

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Partial Derivatives Explained
Relearning calculus is rough
Is there anyone out there who can explain partial derivatives?
Jacobian
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.