To any and all mathematicians

[Anonymous No.8613044]

How tf am I supposed to visualize/conceptualize connection forms?

Things I understand:

>a 1-form, df, is the outside derivative of a real valued function, f: R^n --> R,
>and a k-form is just the outside derivative of a (k-1)-form

>if f: R^3 --> R, then df: R^3 x R^3 --> R
>So, df takes a point and a vector and shows how f varies at the point in the vector's direction

>If you take the covariant derivative of one frame field to another, you get a matrix of 1-forms, called connection forms

So what is special about the (1,2) and (1,3) entries, but not the (2,3) entry? Like, how does this relate to the shape operator?

[Anonymous No.8613863]

>>8613044
I always found it much easier to consider analysis from an algebraic perspective. Don't even bother with visualization and intuition at this level. Just get very good at the logic of symbolic manipulation and then the more clear geometric interpretations will link later once you master the more menial thing

[Anonymous No.8614251]

>>8613044
Explain like I'm 5
>not a 5 yr old

[Anonymous No.8614267]

>>8613044
Don't visualize you idiot.

[Anonymous No.8614277]

>>8613863
>I always found it much easier to consider analysis from an algebraic perspective. Don't even bother with visualization and intuition at this level. Just get very good at the logic of symbolic manipulation and then the more clear geometric interpretations will link later once you master the more menial thing

Wow, I've never heard a more succinct summary of why analysis is the shittiest, ugliest, and least intuitive branch of mathematics. Thanks anon.

t. combinatorialist.

[Anonymous No.8614309]

>>8613863
>Just get very good at the logic of symbolic manipulation

Can't just teach
[math](a*b)^n = a^n*b^n \\ Yet,~if~n=(a^x+1)[/math]
There's a skipping iteration occurring, would you agree?
That shit's meh on abstract-ness

[OHP No.8614332]

>>8614277
This to be quite honest. It seems more natural to gain an intuitive understanding of the machinery driving mathematics on a simple abstract level than it does to just muddle through the rules and let intuition build around those. The former allows you to translate the philosophies of a field to other areas, whereas the latter rigidifies one's understanding and prevents them from moving between ideas fluidly.