Computing a Jacobian matrix

This finishes the introduction of the Jacobian matrix, working out the computations for the example shown in the last video.

[ellingtonkirby]

As someone who is returning to a math intensive program after almost 10 years being out of school, this video has really helped me! Thank you so much

[dalisabe]

Excellent presentation. I love how you made the geometric interpretation of the algebra and calculus involved. I definitely would like an amplified picture of the zoomed area on a separate screen and see exactly what too place after the coordinates change or distortion. Not that it is so important to keep track of minute changes at every point due to such distortion, but it gives a sense of assurance that the concept is well understood. The Jacobian is particularly useful in continuum mechanics where material deformity is studied as in metals and plastic.

[tristanbatchler]

I wish I had this when we did variable transformations last semester :(

[chickenstrangler]

Clicked to get my mid blown, it was great.

[raybroomall]

Pre Calc at 13 is dope... don't get bent.    If you have any idea of Linear Algebra then this isn't too difficult.    But with a full load of Pre Calc  you may not have time for this.   If nothing else, this is telling you why you need Pre Calc .   If you can visualize what he did here the two points that you can take away is that you can plug in any two number pairs into the last matrix (cos) and it will spit out how that point in the XY plane has been changed by the function.  And that any non linear function can be made to look linear if you take a close enough look. (the limit of the function)   It's hard work but you are doing it, and that's cool.

[siddhantgoyal]

So if you multiply the Jacobian matrix with (-2, 1), it should give the same result as if you calculated the (x+sin(y), y+sin(x)) with (-2, 1). Unfortunately, that is not the case. Where am I going wrong here?

[sofisticatedranchbroh]

Does jacobian and hessians only apply to functions going fron R^n space to R^n space?? I first learned Jacobian with regard to R^2 to R^1 space, like traditional 3d functions but this exapmle and general explanations seem to be talking about transformations*** where we have the in with the same dimensionality as the out. Comments? Please!

[GermanSnipe]

So if you zoom in on ANY function is it locally linear? Or just special functions? If only certain ones are locally linear does the Jacobian have any use for them?