Continuity of Multivariable Functions

Welcome to my video series on Multivariable Differential Calculus. You can access the full playlist here:
• Multivariable Differen...
Videos by Zack Cramer, University of Waterloo. Textbook reference: Calculus: Early Transcendentals by Jon Rogawski, Colin Adams, and Robert Franzosa

[@sunraiii]

Great video, and the trick at the end, transforming cartesian to polar coordinates to have one less variable to worry about and approach (0,0) “from all directions”, is really cool!

[@MrCEO-jw1vm]

really good explanation! thanks

[@m.varunreddy7365]

Beautiful video, loved the visuals(graphs,3d), they help me understand faster... thank you suur!

[@reijo6079]

Thank you so much, this really made helped understand how to solve basically any kind of problem like this and also how to make functions continuous in certain points!

[@Stefabro]

you're really good at explaining, thank you for all your help!

[@danieljulian4676]

Nice presentation. First commendation is the figure at the beginning that shows a discontinuity in a surface over the domain R^2, which really brings home the bivalent situation at the discontinuity. You cover the topic (which is a standard one) and include all the usual points, including the transformation to polar coordinates. When you mention applying L'Hôpital's rule, valid on domain R, the series expansion of the function is (at least) implied, and this is also valid in domains beyond R. With the function you chose, the linear term is sufficient to show the limit exists, unless I'm forgetting something important.

[@MiguelSantos-lu8uj]

Great video! Very underated channel with amazing quality. Love from Portugal

[@mathemation]

Thank you, Miguel! So glad you have been enjoying this content :)

[@user-is9os1th3z]

Very helpful video with a very good explanation thank you so much

[@lav6978]

Arigato gozaimuch for the wonderful explanations

[@dzdeparsio4676]

Thank you so much

[@abdelrahmanabdelrahman8149]

you are so good

[@mwilamwamba1787]

Thank you🎉

[@satarnaghizadeh6623]

Awesome ❤

[@ijustwanttoseealiadabbing2270]

love you

[@matrixtrace]

Last example: would it be legal to make a substitution z=x^2+y^2, without transforming to polar coordinates?

[@mathemation]

Yes, that's fine. This works because (x,y)->(0,0) if and only if x^2+y^2 -> 0. Effectively, this is the same as transforming to polar coordinates.

[@isaacmarcelino_]

How do we know the best paths to take?

[@rogersowden2369]

Hey i checked your video before this but I couldn't see anything about x^2 + y^2 = p^2? where do I find these cartesian to polar conversions?

[@mathemation]

The conversion between Cartesian and polar coordinates are discussed here: czcams.com/video/_MXTd_ZUKFg/video.html

[@rogersowden2369]

@@mathemation Thanks for the quick reply! You're awesome! Subbed

[@shams8917]

💛