Find Exterior Derivative of Differential Forms in Dim > 3

[saminator910]

So say I have a n-1 form

\sum^{n}_{i=1}x^{2}_{i}dx_{1}...\widehat{dx_{i}}...dx_{n}

and I want to find the exterior derivative, how do I know where to put which partial derivative for each term,

would it simply be??

\sum^{n}_{i=1} \frac{∂x^{2}_{i}}{∂x_{i}}dx_{i}dx_{1}...\widehat{dx_{i}}...dx_{n}

hopefully this will clarify, for this 2-form

\alpha=x_{1}x_{2}dx_{2}dx_{4} + x_{3}x_{4}dx_{1}dx_{2}

how would one go about finding the exterior derivative? I have no idea which partials to put where, this is simple for a normal ℝ³ 2 form, but I have no idea here. let me know if I need to clarify.

 

[tiny-tim]

hi saminator910!

i'll rewrite your question slightly, since i find it a little confusing …

how do i find the exterior derivative of an n-1 form \sum^{n}_{i=1}f_i(x_1,\cdots x_n)dx_{1}..\cdots\widehat{dx_{i}}\cdots dx_{n} ?​

i find it easier to write it as (\sum^{n}_{l=1}d_j)\wedge\sum^{n}_{i=1}f_i(x_1,...x_n)dx_{1}\wedge\cdots \widehat{dx_{i}}\cdots\wedge dx_{n}

then everything except j = i is zero, and you get

(\sum^{n}_{i=1}\partial f_i(x_1,...x_n)/\partial x_i)\ \ dx_{1}\wedge\cdots \wedge dx_{n}

hopefully this will clarify, for this 2-form

\alpha=x_{1}x_{2}dx_{2}dx_{4} + x_{3}x_{4}dx_{1}dx_{2}

how would one go about finding the exterior derivative? I have no idea which partials to put where, this is simple for a normal ℝ³ 2 form, but I have no idea here. let me know if I need to clarify.

if it's in ℝ³, where does x₄ come from?

 

[saminator910]

thanks a lot, this seems to make sense. That is actually in ℝ⁴, how would one solve that? I was saying it would be simple for a 2-form in ℝ³, but it's difficult in ℝ⁴.

 

[tiny-tim]

ok, then eg d\wedge (x_1x_2\,dx_2\wedge dx_4)

= \partial (x_1x_2)/\partial x_1\ (dx_1\wedge dx_2\wedge dx_4)

+ \partial (x_1x_2)/\partial x_3\ (dx_3\wedge dx_2\wedge dx_4)

= x_2\ (dx_1\wedge dx_2\wedge dx_4)

 

[saminator910]

Okay thanks a lot , that really makes things clearer. So for the 2 form in ℝ^{4}

I'm going through this step by step, just in case I make a mistake...

β=x_{1}x_{2}dx_{3}dx_{4}+x_{3}x_{4}dx_{1}dx_{2}

dβ=d(x_{1}x_{2})dx_{3}dx_{4}+d(x_{3}x_{4})dx_{1}dx_{2}

dβ=(x_{2}dx_{1}+x_{1}dx_{2})dx_{3}dx_{4}+(x_{4}dx_{3}+x_{3}dx_{4})dx_{1}dx_{2}

now from here is where I think I'm doing something wrong, I "distribute" if you will, the dx's outside the parenthesis and get

dβ=x_{2}dx_{1}dx_{3}dx_{4}+x_{1}dx_{2}dx_{3}dx_{4}+x_{4}dx_{3}dx_{1}dx_{2}+x_{3}dx_{4}dx_{1}dx_{2}

but the supposed answer is this, notice the switched dx's in the last two terms, why do I need to do this?

dβ=x_{2}dx_{1}dx_{3}dx_{4}+x_{1}dx_{2}dx_{3}dx_{4}+x_{4}dx_{1}dx_{2}dx_{3}+x_{3}dx_{1}dx_{2}dx_{4}

 

[tiny-tim]

hi saminator910!

β=x_{1}x_{2}dx_{3}dx_{4}+x_{3}x_{4}dx_{1}dx_{2}
…dβ=x_{2}dx_{1}dx_{3}dx_{4}+x_{1}dx_{2}dx_{3}dx_{4}+x_{4}dx_{3}dx_{1}dx_{2}+x_{3}dx_{4}dx_{1}dx_{2}

that's right

but the supposed answer is this, notice the switched dx's in the last two terms, why do I need to do this?

dβ=x_{2}dx_{1}dx_{3}dx_{4}+x_{1}dx_{2}dx_{3}dx_{4}+x_{4}dx_{1}dx_{2}dx_{3}+x_{3}dx_{1}dx_{2}dx_{4}

you don't need to do it, it's just neater

$dx_{3}\wedge dx_{1}\wedge dx_{2}$ is the same as $dx_{1}\wedge dx_{2}\wedge dx_{3}$ (it's an even number of exchanges, so there's no minus-one factor)

but the latter looks better ​

 

[saminator910]

Thanks a lot!

 

[HallsofIvy]

hopefully this will clarify, for this 4-form

\alpha=x_{1}x_{2}dx_{2}dx_{4} + x_{3}x_{4}dx_{1}dx_{2}

how would one go about finding the exterior derivative? I have no idea which partials to put where, this is simple for a normal ℝ³ 2 form, but I have no idea here. let me know if I need to clarify.

Its differential is d\alpha= d(x_{1}x_{2}dx_{2}dx_{4} + x_{3}x_{4}dx_{1}dx_{2})
= (x_2dx_1+ x_1dx_2)(dx_2dx_4)+ (x_4dx_3+ x_3dx_4)(dx_1dx_2)
= (x_2dx_1dx_2dx_4+ x_1dx_2dx_2dx_4)+ (x_4dx_3dx_1dx_2+ x_3dx_4dx_1dx_2)
Now use the fact that the multiplication is "anti-symmetric" (which immediately implies that dx_2dx_2= 0) to write that as
x_2dx_1dx_2dx_4+ x_4dx_1dx_2dx3+ x_3dx_1d_2d_3

The first term, x_2dx_1dx_2dx_4 already has the differentials in the "correct" order. The last two, x_4dx_3dx_1dx_2 and x_3dx_4dx_1dx_2, each require two transpositions, x_4dx_3dx_1dx_2 to -x_4d_1dx_3dx_2 to -(-x_4dx_1dx_2dx_3) and x_3dx_4dx_1dx_2 to -x_3dx_1dx_4dx_2 to -(-x_3dx_1dx_2dx_4) and so have no net change in sign.

(What is the "correct" order is, of course, purely conventional.)