Can someone explain (Differential Forms)

In summary: The notation i < j is just to avoid summing over pairs (i, j) with i = j, because then αi ∧ βi = βi ∧ αi = 0.
  • #1
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(i) if [itex]\alpha=\sum_i \alpha_i(x) dx_i \in \Omega^1, \beta=\sum_j \beta_j(x) dx_j[/itex]

then[itex]\alpha \wedge \beta = \sum_{i,j} \alpha_i(x) \beta_j(x) dx_i \wdge dx_j \in \Omega^2[/itex]

NOW THE STEP I DON'T FOLLOW - he jumps to this in the lecture notes:

[itex]\alpha \wedge \beta = \sum_{i<j} (\alpha_i \beta_j - \beta_j \alpha_i) dx_i \wedge dx_j[/itex]

the subscript on the sum was either i<j or i,j - could someone tell me which as well as explaing where on Earth this step comes from.

(ii) could someone explain the Leibniz rule for exterior derivative [itex]d: \Omega^k \rightarrow \Omega^{k+1}[/itex]

i.e. why [itex]d(\alpha^k \wedge \beta^l)=d \alpha^k \wedge \beta^l + (-1)^k \alpha^k \wedge d \beta^l[/itex]
note that the superscript on the differential form indicates that it's a k form or and l form

my main problem here is where the (-1)^k comes from

cheers for your help
 
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  • #2
Hi latentcorpse! :smile:
latentcorpse said:
[itex]\alpha \wedge \beta = \sum_{i<j} (\alpha_i \beta_j - \beta_j \alpha_i) dx_i \wedge dx_j[/itex]

the subscript on the sum was either i<j or i,j - could someone tell me which as well as explaing where on Earth this step comes from.

It's i < j …

[itex]\alpha \wedge \beta = \sum_{i,j} (\alpha_i \beta_j) dx_i \wedge dx_j = \sum_{i<j} (\alpha_i \beta_j - \beta_j \alpha_i) dx_i \wedge dx_j[/itex]

since wedge is anti-commutative.
why [itex]d(\alpha^k \wedge \beta^l)=d \alpha^k \wedge \beta^l + (-1)^k \alpha^k \wedge d \beta^l[/itex]

my main problem here is where the (-1)^k comes from

d() is like another wedge, so if you move d() through k elementary forms, you multiply it by (-1)k :wink:
 
  • #3
cool i get the 2nd part now - still not sure why it's i<j or why [itex]\alpha_i \beta_j[/itex] suddenly becomes [itex]\alpha_i \beta_j-\beta_j \alpha_i[/itex]
 
  • #4
(have an alpha: α and a beta: β and a wedge: ∧ :wink:)
latentcorpse said:
… still not sure why it's i<j or why [itex]\alpha_i \beta_j[/itex] suddenly becomes [itex]\alpha_i \beta_j-\beta_j \alpha_i[/itex]

because αi ∧ βj = -βj ∧ αi

so if you sum αi ∧ βj over all i and j,

then for i > j you just "turn it round" :wink:
 

1. What are differential forms?

Differential forms are mathematical objects used in multivariable calculus to describe and analyze geometric properties of surfaces and volumes. They are represented by expressions involving coordinate differentials and can be thought of as "generalized functions" that can be integrated over a particular region.

2. How are differential forms different from traditional vector calculus?

Differential forms differ from traditional vector calculus in that they are defined independently of any coordinate system. This allows them to be used in a more general and coordinate-independent manner, making them particularly useful in abstract and higher-dimensional spaces.

3. What are some practical applications of differential forms?

Differential forms have a wide range of applications in various fields such as physics, engineering, and computer graphics. They are used to solve problems involving fluid mechanics, electromagnetism, and optimization, among others. They can also be used to study the geometry of curves, surfaces, and higher-dimensional objects.

4. What are the fundamental operations on differential forms?

The fundamental operations on differential forms are exterior derivative, interior product, and wedge product. The exterior derivative is a generalization of the gradient operator and measures how a form changes over a small region. The interior product and wedge product are used to combine and manipulate forms, similar to how dot and cross products are used in traditional vector calculus.

5. Are there any prerequisites for understanding differential forms?

A basic understanding of multivariable calculus and linear algebra is necessary to understand differential forms. Familiarity with concepts such as gradients, partial derivatives, and determinants is helpful in grasping the fundamentals of differential forms. Some background in abstract algebra and topology may also be beneficial, but is not required.

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