Local linearity for a multivariable function

[@AuroraNora3]

3blue1brown's animations are amazing

[@elbay2]

Indeed!

[@elbay2]

Question: are all functions that are made up of various combinations of elementary functions locally linear, or are there some that are not?

[@whupdup62]

It looks to me that so long is a function is continuous it will be locally linear, and I can't think of a counterexample off the top of my head.

[@elbay2]

Thanks!

[@ozzyfromspace]

Counter-example: |x| is continuous at x=0, but not differentiable at that point. +elbay2, A better condition for what you described is that the function must have well-defined partial derivatives at that point (which |x| has everywhere except x=0). This is kind of my first time really seeing this locally linear stuff, so I don't really have any examples in mind; just the concept of differentiability. A hint for this viewpoint is found in the Jacobian matrix, which is built up of partials. I hope this helps. Sorry if I got anything wrong #NewLearner. -Float Circuit.

[@lukemoura256]

@@ozzyfromspace if you were to play the animation of the transformation of |x| every positive input would map to itself which means this function is locally linear at all regions that don't include negative values

[@douglasmangini8744]

@@lukemoura256 actually it would be locally linear at every point except for zero. That's because the condition for local linearity is differentiability. And a sufficient condition for differentiability is the existence of continuous partial derivatives.

[@LoveLauwsPawuwls]

At my university, we have to take multi variable calc before linear algebra so I'm sort of unfamiliar with this explanation

[@harry_page]

I don't know if you wanted this explanation but if you do: linear algebra deals with fully linear transformations, which are just the small subset of these vector to vector functions that look like f(xi + yj) = (ax + by)i + (cx + dy)j where a,b,c,d are all just constants. These transformations keep all the gridlines straight and evenly spaced globally, throughout the space, not just when you zoom in on a point

[@bosepukur]

such amazing content

[@falnica]

Can you please share the program you use for the transformations?

[@jonasdaverio9369]

Fernando Franco Félix www.3blue1brown.com/about/

[@austinfogleman474]

Its called manim

[@Rabbit-el1cr]

This should have more views.

[@ankykolo6526]

Thank You Sir! Could you pleeeeeease metion the name of the program you used for this transformations ??

[@shawaizhaider3978]

manim. funfact: Its a python library for math animation written by him.

[@yazicib1]

Is there a way to find out what playlist this video is part of? I want to start from the beginning... at least previous video?

[@primarygarbage]

kzbin.info/aero/PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7

[@fernandointriagoveliz6290]

The hero we need but don't deserve

[@rafa_beirigo]

Just awesome, thanks!

[@4ur3n]

what program is he using to show all that?

[@saikumartadi8494]

hail 3 blue 1 brown!!

[@GOODBOY-vt1cf]

thank you so much

[@turboleggy]

Thank you!

[@陈昱翔-n6l]

amazing!!!

[@paulcalibur]

WOW