Could you help me see my mistake?

Artemisia

14:48

Greetings Sir,

I am so sorry to impose... if I did

Could you spare some time to help me in a question in which I can't really find my mistake?

I really don't mean to be bothersome

Pablo Rotondo

Hello

Artemisia

Hi :)

I am so sorry if I am bothering you

Pablo Rotondo

No problem, as long as it is not a really long question.

xD

Artemisia

I have a question... I have asked a few people for feedback and I took their advice... but the answers are apparently incorrect

Pablo Rotondo

Okay

Artemisia

14:53

0

Q: Continuity, differentiability and existence of partial derivatives

Here are a few functions whose continuity, differentiability and existence of partial derivatives are to be checked at the origin. I have given the answers, but I would really appreciate it if someone could check it for me :) $$1. f(x,y)=\sin x\sin(x+y)\sin(x-y)$$ Continuous, differentiable, part...

multivariable-calculus

This is actually an online assignment... I am new to these concepts and I seem to be struggling with them

And I have to enter these answers in a multiple choice format... however, I have tried asking around and most people say this is right, but the system doesn't accept it

And I can't point any errors out

Haha I just need a once over :) It isn't too long, I hope :)

Pablo Rotondo

I think

the problem is #6

because actually

Artemisia

Ok...

Pablo Rotondo

the partial derivatives

do not seem to exist

let me think a bit

Artemisia

Ok sure :)

Pablo Rotondo

the problem is that

$\sqrt{h^2} = |h|$

not $h$

Artemisia

14:58

ok...

Pablo Rotondo

And so, the limit depends on whether

you come from left

or right

you get either -1 or 1

Artemisia

Oh I see

That means the partial derivatives don't exist

Pablo Rotondo

exact;y

Artemisia

which also implies that the function is not differentiable?

Pablo Rotondo

yes

Artemisia

15:00

But it is continuous because it is trigonometric

Pablo Rotondo

indeed, you have compositions and sums of continuous functions

Artemisia

Oh I get it now :) but are the others fine?

Pablo Rotondo

wait a minute, I will look at them again

Artemisia

Thank you :)

Pablo Rotondo

in 3

you have the same issue

with the partial derivatives

see?

Artemisia

15:02

Oh but this is about the origin...

Pablo Rotondo

right

Artemisia

Aren't the partial derivatives zero?

Pablo Rotondo

ah, sorry

haha

I was thinking about $|x|+|y|$

indeed you are right

Artemisia

haha :)

Pablo Rotondo

it is 0 on both axis

Artemisia

15:04

So, other than 6, all seem right?

Oh all the functions ask for differentiability and continuity on the origin :)

Pablo Rotondo

still checking

in 4 the partial derivatives are undefined indeed, but $f(0,0)=0$ by definition, not $0/0$.

Artemisia

Oh but I thought we are approaching 0. So we choose the other function at 0+h

Pablo Rotondo

Yes, you get $1/h$ with $h\to 0$ in the case of $\partial_x$, and indeed, that is not defined

but, it is not because of that $0/0$, since $f(0,0)$ was forcefully defined actually

Artemisia

OH! Haha I understand now :)

Pablo Rotondo

Okay, it seems good

Artemisia

15:08

I have another question, but this is more general

Pablo Rotondo

Yes

Artemisia

I have an exam on groups and symmetries around the corner (but no rings)

Pablo Rotondo

Okay

Artemisia

do you have any pointers that will help me?

in the course of my preparation

I have always had a problem with abstract algebra

Pablo Rotondo

Mmmmmm, I am not sure if there is some sort of advice I could give you, the thing is, you should try to appreciate the beauty of it I think

at least, group theory, is really beautiful

Artemisia

15:10

yes it is :) but my lectures are so mathematical that we don't really understand any of the concepts

it's just pages of proofs

Pablo Rotondo

I see... then maybe you could try doing some problems to get a feel for it

in a way, you could say it is a lot about little "puzzles"

Artemisia

I did try haha, but I quickly lost interest

Yes :)

I like the 15-puzzle solution, got hold of a board and tried so hard to get it right

Pablo Rotondo

Hahaha

Artemisia

Haha and then I realized all my attempts are futile

So disappointing :)

but I understood the theory behind it

Thank you so much for all your help :)

I really appreciate it

Pablo Rotondo

You are welcome

Artemisia

15:13

If I have some queries along the way, may I approach you?

Pablo Rotondo

hahahahaha

With group theory?

Artemisia

haha yeah :)

Pablo Rotondo

Alright

Hahaha

Artemisia

or multivariable calculus

:) Thank you so much :) Are you a professor?

Pablo Rotondo

Haha, no, I am a student.

Artemisia

15:14

Oh wow :) PhD?

Pablo Rotondo

Not yet, most majors are quite long in my country, and I am double majoring maths and Computer Science

Artemisia

Oh ok :) I am doing a double major in Physics and Math

Minors in Philosophy and Psychology

Pablo Rotondo

Hehe, it's been a while since I last studied physics

xD

Artemisia

Hahaha :)

For me, Math is more of a supplement

I truly enjoy Physics :)

:) Once again, thank you so much for all your help :)

I shall get back to frieze groups :)

Pablo Rotondo

You are welcome.

Good luck

Artemisia

15:17

Thank you :) Good luck to you too :)

Transcript for

$Mathematics$ Could you help me see my mistake?