Multivariable Calculus | The Jacobian

With the ultimate goal of building a change of variables formula for multiple integrals, we motivate the Jacobian and give some example calculations.
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[@liviumircea6905]

This is a gem...like other people said this is the best content about Jacobian I found...Nice balance between intuition and formalism.Thank you very much Michael Penn

[@azmath2059]

Finally! A proper and concise explanation of the Jacobian! Thanks Michael.

[@pendawarrior]

Best explanation of the jacobian I've seen, thank you so much

[@thenateman27]

Mistake at 14:46, it should be a positive 2w

[@Hobbit183]

Really nice explanation, thanks

[@Sidd-rb4ec]

At 3:44 i don't understand why did we take tangent vector for the approximation

[@danv8718]

Because the tangent line always gives you the best approximation to a curve at a given point, in fact, if you zoom close enough, both will be indistinguishable. Zooming close enough is exactly what we do when we take the limit as the variables tend to zero and therefore the area of the rectangle and the area of the curvilinear region will match. Bottom line is all curvilinear lines look like straight lines if you get close enough.

[@guvencagil]

Excellent. Thanks.

[@antoniocosta1447]

The final result should be 2w × (u×exp(2v) - u×exp(v + w + vw) )

[@programstix3837]

thank you

[@РоманЛева-й9з]

Спасибо, хоть кто-то внятно проводит доказательства

[@tsunningwah3471]

diu

[@ANTOINETTE-nk1tm]

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[@ANTOINETTE-nk1tm]

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