Oxford Calculus: Classifying 2D Critical Points using the Discriminant

[@TomRocksMaths]

Part 1 on how to find critical points here: kzbin.info/www/bejne/gpbSnqirbZdgq5o

[@jamesgreenwood6997]

This discriminant looks suspiciously like the determinant of the Hessian matrix. Is the determinant of the Hessian Matrix used to classify critical points for functions of three or more variables? What if the discriminant is equal to zero?

[@srinandanasastry3001]

Sir,you just explained the whole thing in half an hour, what I was trying to understand for the past one month.... Awesome video 🤩🤩,I will be eagerly waiting more such videos 😊😄

[@TomRocksMaths]

Thanks Srinandana - glad it helped!

[@liamwhite7890]

Have been trying to understand this for a good few weeks and you have just given the best explanation and proof . Thank you 🙏🏽

[@TomRocksMaths]

Glad it was helpful Liam!

[@zulfiqarjafri1492]

Great explanation. Absolutely awesome!

[@fahimahmedbhuiyan2252]

Always hit the like button even before I start to watch...

[@TomRocksMaths]

[@jclopez5447]

Man, you are a legend 💯

[@TomRocksMaths]

Thanks JC :)

[@amitbenjam]

Beautiful concept, most often learned technically but beyond that it emphasises the recursive behavior of differentiation in different dimensions The theory is not rewritten, but applied in different forms, keeping this in mind I feel helped me overcome the overwhelming equations and fresh material when I started multivariable calculus

[@MathRocks]

Cheers my friend, great job

[@TomRocksMaths]

Thanks Marcos :)

[@PapaFlammy69]

:)

[@louiswong921]

papa flammy is an element of the smart people

[@adamrjhughes]

18:50 what if the discriminant was negative, however the right hand product within the brackets was less than the left hand product with in the brackets so that the sum in the brackets was positive, thus the whole Right hand side was negative? Is it that this case is impossible whats going on what have I missed. Love the videos 👍

[@adamrjhughes]

Ahhhhhh it needs to be always positive to describe the bowl shape

[@TomRocksMaths]

Yes, if the discriminant is negative then the sign of the RHS can be either positive or negative depending on which direction (or line) you are moving in. This is why it has to be a saddle, because it's not a max or min, and we know we are at a critical point, which leaves the saddle as the only option.

[@adamrjhughes]

@@TomRocksMaths Thanks for the reply! When this clicked, I felt so stupid. Great vids!

[@TomRocksMaths]

Don't feel stupid, feel glad that you figured it out :)

[@riddhimanna8437]

You're an awesome teacher!

[@TomRocksMaths]

Thank you Riddhi! 😃

[@jorgelechon8044]

Thanks for another excellent video.

[@TomRocksMaths]

Thanks Jorge - glad you enjoyed it!

[@Unreql]

Is it just me or do the Maplesoft worksheets not work? They just direct you to the home page.

[@waiswarichard9339]

i loved the lesson, thank you so much. i have a qn how do you conclude when both fxx and fyy = 0 and D< 0

[@TomRocksMaths]

If D

[@fahrenheit2101]

I get that the second order partial x derivative represents the rate of change of gradient in the x direction, and ao does the second order partial y derivative, but what does the second order partial x derivative followed by partial y derivative represent? Is it another direction?

[@TomRocksMaths]

It's the change in the y-direction of the x-derivative

[@sardarbekomurbekov1030]

Thank you

[@TomRocksMaths]

You're very welcome Sardarbek.

[@PixelSergey]

What if the discriminant is negative but small? Wouldn't the first squared term "trump" over the sum, leaving the sign dependent only on fₓₓ?

[@TomRocksMaths]

You have to be careful, because we can always just make y-y0 really large and then the square bracket becomes negative. If y-y0 is small too, then the square bracket will indeed be positive as you say.

[@eliecerecology]

Awesomeness!

[@TomRocksMaths]

Glad you enjoyed it :)

[@kenana3456]

Great video, I focused more on your good chalks Are you a chalks dealer😂😂😂? just kidding.

[@alexeyantoshin3433]

Отличное видео )))

[@LongArmsShortTorse]

Hi Tom. really enjoyed this video but I have some questions. Please could you tell me if the maple worksheet still exists. I can't seem to access it though you're commentary and notes were definitely very easy to follow between 15:04 and 15:26. There was a 3blue1brown video on Taylor series and he was using it to approximate points using polynomials to approximate points of other functions by differentiating the function and comparing coefficients. Is this a different use of Taylor series as it doesn't seem to resonate with his video? kzbin.info/www/bejne/aZWZdaagfqetgJY Sorry for being patronizing, this isn't my intention. I'd just like to know. I'll watch you're other Taylor series videos as I think they'll give me a better understanding. Thank you for producing this epic video.

[@tavishu]

What if the discriminant is zero?

[@TomRocksMaths]

Good question, and correctly answered by Likith Magnet - thanks!

[@My-Say]

Tom, I'm not real great at math beyond algebra, so is there a physics explanation why scientists are making the bizarre claim the universe is expanding when atoms are not? know the physics involving QM is apparently different than standard physics, but the idea the universe is expanding seems to be based entirely in the res shift phenomenon, but isn't the tired light theory more reasonable? And if it somehow does get bigger, it much have already been bigger. Maybe I understand the universe wrong. I always understood it as everything that exists. This would include the nothingness of space, which is still something. It isn't like there's a wall at the edge of the universe that simply expands like a perfect bubble in all directions as a speck of dust moves "outside of it". No, I think the universe is infinite in age and size, and the CMBR waves are from a black hole explosion, and they happen all the time. I would love to understand the math required to test my ideas on all this, but think I need to start from more basic math and work my way up to advanced physics.

[@TomRocksMaths]

Hi Steve, this is definitely beyond my expertise, but I believe the discovery of gravitational eaves also adds quite a lot of evidence to support the expansion of the universe. Might be worth checking out some of stuff around that topic?

[@son_et_lumiere9]

what happens when the determinant is equal to zero?

[@TomRocksMaths]

That's a tricky case where you have to use other methods of investigation unfortunately.

[@DeadJDona]

🍄

[@danielm.1441]

Those are some girthy chalks...

[@TomRocksMaths]

Bigger is better no?

[@danielm.1441]

@@TomRocksMaths Always.