What is differentiability for multivariable functions??

How should we define differentiability of multivaraible functions? We saw in the previous video of our Calc III playlist that partial derivatives is not enough because we saw an example of a discontinuous function whose partials didn't exist, violating our intuition from Calculus I. In this video we will dive back into Calc 1 and rephrase the normal limit definition of the derivative in terms of errors. That formulation of the derivative concept extends nicely to the multivariable case and gives us a nice definition. Finally we note the theorem that if the partials exist and are continuous then we have existence of the derivative.

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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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[DrTrefor]

This video actually defines differentiability too strongly. I introduced two error functions. However, I incorrectly said those error functions were only functions of x and y respectively. Both should be functions of BOTH variables and small when the limit as BOTH variables goes to zero.

[dAntony]

I don't understand why this 15 minute video is more clear and concise than my 1 hour 50 minute lecture that occurs 2xs per week.

[ritxve]

Wow! best explanation of differentiability in multivariable calculus, I have never seen someone explaining single variable derivative
in that E(h) form, this helped me to relate quickly E1 & E2 terms.
Thank u very much professor🙏

[ozgeylmaz]

This video was amazing, it brought tears to my eyes! Now, I can view the world from various perspectives thanks to what I've learned.

[abmohit]

outstanding and amazing ... the reason why i love calculus ...and the reason many don't love calculus becuz we don't have the teachers that can show the real beauty of calculus in this way...Really appreciated and very happy that i got someone who teaches the way i want to learn the beautiful things ...such as calculus !!! lots of love sr ...✨✨

[punditgi]

It would be nice to do the epsilon-delta proof that limit exists for the difference quotient for the single variable case, then do the same for the alternate formula with an error term, then do the same for the multi variable function. Finally, prove the differentiable theorem for multi variable functions with continuous partial derivatives.

[giacomogavelli]

Thank you for this video, it gave me an explication to a problem I found doing complex analisys and that's really cool

[praveshkumar]

This is one of the easiest explanation i have seen for this. Thank you

[julioreyram]

Nice intuitions and visualizations. Thanks!

[matanshtepel]

great video! will share with my classmates 😁👍

[amansingh-wwqc]

No one can teach like u sir, love u sir

[Reptilian.cricket]

really great video and whole playlist!!! Thank you :)

[_mippi_]

At 9:23-9:26: small typo for the error term for y. You have E_1 and E_2 as different functions. Not sure if this mattered but I understand what is going on. Fantastic video! :)

[AkhileshYadav-oljy]

A tremendous explanation
Thank you sir
Love ❤️ form india

[abdulghanialmasri]

Another great video, many thanks.
Would you consider more topics in numerical analysis, like stability, interpolation, extrapolation, gaussian quadrature, etc.

[lolololxd]

sos un CAPO increible maestro muchas gracias

[douglas]

Thinking about the derivative when walking along a diagonal path between the X axis and the Y axis. Example (using a large step to be clear, but in reality thinking about a much smaller one), from (0, 0) -> (1, 1).
In the video, you decomposed the resulting derivative as if we were walking a step in the X direction, and then a step on the Y direction.
You then added the partials and the respective errors.
If we first move in the X direction from (0, 0) -> (1, 0), we add the partial X at (0, 0) * delta X, with the error caused by delta X.
And when moving from (1, 0) -> (1, 1), we add the partial Y at (0, 0) * delta Y, with the error caused by delta Y.
But in the second step, since we are now moving from (1, 0) and not from (0, 0), shouldn't the Y derivative not be evaluated at (0, 0), but at (1, 0)?
In other words, dF/dy on the equation at 10:38 should be evaluated at the point (X0 + delta X, Y0), and not (X0, Y0)?

[SanjinHalilhodzic-nyzj]

Great video, just one question. What is considered small for the error terms?

[Bermatematika]

Usually to define differentiability for function of two variables, I try to convey it geometrically. I say that analogous to the one dimensional case, to be differentiable meant that we want to have a plane that approximate the surface "well enough". How we define this notion of well enough? If we move (x, y) to (x+delta x, y+delta y) the error of the z value between the surface and the plane the relative error of the z values between the surface and the plane with respect to the displacement from (x, y) to (x, + delta x, y+ delta y) is goes to zero.

[Mark-dcsu]

This kind of reminds me of the squeeze theorem in a way. The assumption being that if something has a partial derivative in the x and y directions, there should be a point between the x and y axis where the function is defined. Is this a valid way of understanding differentiability?