Why the terms - exterior, closed, exact?

An exact form is always closed because the boundary of a boundary is always zero. In other words, the image of a differential operator applied to a differential form is always a zero differential form.
  • #1
82
11
Hi all,

(Thank you for the continuing responses to my other questions...)

I am gaining more and more understanding of differential forms and differential geometry.

But now I must ask... Why the words?

I understand the exterior derivative, but why is it called "exterior?"
Ditto for CLOSED and EXACT forms: I understand the definition, but nothing in that definition informs me of why that word was chosen.

Could anyone elaborate, simply, on the reason for the words: exterior, closed, exact?
 
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  • #2
observer1 said:
Hi all,

(Thank you for the continuing responses to my other questions...)

I am gaining more and more understanding of differential forms and differential geometry.

But now I must ask... Why the words?

I understand the exterior derivative, but why is it called "exterior?"
Ditto for CLOSED and EXACT forms: I understand the definition, but nothing in that definition informs me of why that word was chosen.

Could anyone elaborate, simply, on the reason for the words: exterior, closed, exact?
This comes from its topological, resp. geometrical origin. A closed differential form is a cycle ##C##, a exact differential form a boundary ##B## which establishes the short exact sequence: ##0 → B → C → C/B → 0##. (A exact form is always closed.) See also Wiki: "Loosely speaking, a cycle is a closed submanifold, a boundary is the boundary of a submanifold with boundary, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries." Geometrically it is related to simplicial complexes where the language comes from.

Edit: I omitted the filtration for simplicity.

Edit 2: The Graßmann algebra ##Λ V## is also called the exterior algebra of ##V##. It is the structure where exterior derivatives live. It is exterior because a inner product on ##V## would make it an algebra with a binary multiplication ##V \times V → V## whereas the exterior product is between the filtration of ##Λ V##, e.g. ##a ∧ b = (-1)^{ij} b ∧ a## for ##a \in Λ^i(V)## and ##b \in Λ^j(V)##.
 
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  • #3
fresh_42 said:
This comes from its topological, resp. geometrical origin. A closed differential form is a cycle ##C##, a exact differential form a boundary ##B## which establishes the short exact sequence: ##0 → B → C → C/B → 0##. (A exact form is always closed.) See also Wiki: "Loosely speaking, a cycle is a closed submanifold, a boundary is the boundary of a submanifold with boundary, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries." Geometrically it is related to simplicial complexes where the language comes from.

Edit: I omitted the filtration for simplicity.

Edit 2: The Graßmann algebra ##Λ V## is also called the exterior algebra of ##V##. It is the structure where exterior derivatives live. It is exterior because a inner product on ##V## would make it an algebra with a binary multiplication ##V \times V → V## whereas the exterior product is between the filtration of ##Λ V##, e.g. ##a ∧ b = (-1)^{ij} b ∧ a## for ##a \in Λ^i(V)## and ##b \in Λ^j(V)##.

I am still learning. Almost at the point of understanding some of what you wrote.

But could you try that explanation again.. but simpler... even if you have to exaggerate... I am looking for a picture of the idea/definition.

In other words: these terms are often defined right at the start of an introduction to differential forms...
That means if they are using a word (exterior, closed, open) right there at the start, they should give a reason for the word, based on the minimal knowledge a first time reader has.

You have given me the reason one would give AFTER understanding it.
 
  • #5
observer1 said:
But could you try that explanation again.. but simpler... even if you have to exaggerate... I am looking for a picture of the idea/definition.
Well, I can give it a try although the explanation above has been already rather simplified.

exterior
An algebra is a vector space ##\mathcal{A}## which allows a (interior) multiplication, e.g. ##\mathcal{A} = \{## analytic functions on ##ℂ\}## with ##f\circ g: z \mapsto f(g(z))##.
The Graßmann or exterior algebra ##\Lambda(\mathcal{A}) ## of ##\mathcal{A}## is a vector space of countably many copies of ##\mathcal{A}## together with certain rules (identifications). The exterior multiplication of ##\mathcal{A}## is now the (interior) multiplication in ##\Lambda(\mathcal{A})##. It is between, e.g. an element in ##n## copies of ##\mathcal{A}## with an element in ##m## copies of ##\mathcal{A}##. A differential form can be defined as a alternating, multilinear mapping from such an exterior algebra to the space of smooth functions. Let me quote an example from Wikipedia (translated by me):

Consider ##ℝ^3## with cartesian coordinates ##(x,y,z)##, the 1-form ##ω = z^2 dx +2ydy+xzdz## and the 2-form ##ν=zdz ∧ dx##.
The exterior product is thus
##ω ∧ ν = z^3 dx ∧ dz ∧ dx + 2yz dy ∧ dz ∧ dx + xz^2 dz ∧ dz∧ dx = 2yz dx ∧ dy ∧ dz##
where I applied some commutation rules mentioned above
and the exterior derivation is
\begin{align*}
dω &= d(z^2 dx +2ydy+xzdz) \\
&= (\frac{\partial}{\partial x} z^2 dx + \frac{\partial}{\partial y} z^2 dy + \frac{\partial}{\partial z} z^2 dz) ∧ dx + (\frac{\partial}{\partial x} 2y dx + \frac{\partial}{\partial y} 2y dy + \frac{\partial}{\partial z} 2y dz) ∧ dy + (\frac{\partial}{\partial x} xz dx + \frac{\partial}{\partial y} xz dy + \frac{\partial}{\partial z} xz dz) ∧ dz \\
&= 2z dz ∧ dx + 2 dy ∧ dy + (z dx + xdz) ∧ dz \\
&= 2z dz ∧ dx - z dz ∧ dx \\ &= z dz ∧ dx \\& =ν
\end{align*}

##ν## is exact because it is the image of ##ω## under the boundary, resp. differential operator ##d##.
##ν## is also closed because ##dν=0##.

exact (##ν = dω##)
Boundary operators ##d## always satisfy ##d^2 = d \circ d = 0##. Therefore it's always ## \mathcal{im} \, d ⊆ \mathcal{ker}\, d## and ##\{0\} → \mathcal{im} \, d \stackrel{φ}{→} \mathcal{ker}\, d \stackrel{ψ}{→} \mathcal{ker}\, d / \mathcal{im} \, d → \{0\} ## is a short exact sequence, i.e. ##\mathcal{im} \, φ = \mathcal{ker} \, ψ##. (I don't know why they first called such a sequence exact. Perhaps because it is not only ##\mathcal{im} \, φ ⊆ \mathcal{ker} \, ψ## but exactly ##\mathcal{im} \, φ = \mathcal{ker} \, ψ## which is an important property of such sequences.) You could say as well that a exact form is a boundary (of another form).

(... continued example. Only to show how those abstract concepts fit in situations we know.)
Let ##γ: [0,1] → ℝ^3## be a parametrization ##γ(t) = (t^2,2t,1)## of a curve in ##ℝ^3##.
Thus we have for ##x = t^2 \; , \; y=2t \; , \; z=1## for the pullback ##γ^*##
\begin{align*}
γ^*ω &= ω(dγ)\\ &= (z^2 dx + 2y dy + xz dz)dγ\\
& = (1^2dx + 2\cdot 2t dy + t^2 \cdot 1 dz)dγ\\& =\left(1^2 \frac{d(t^2)}{dt} + 2(2t) \frac{d(2t)}{dt} + t^2 \cdot 1 \frac{d(1)}{dt}\right)\,dt \\&= 2t dt + 4t \cdot 2 dt + 0 \\&= 10t \, dt
\end{align*}
For the integral of ##ω## along the curve ##Γ = γ([0,1]) ⊆ ℝ^3## we thus get ##\int_Γ ω = \int_{[0,1]} \gamma^*ω = \int_0^1 10t \, dt = 5##.

closed (##dν = 0##)
Let ##M## be a compact differential manifold. Let further ##M## be closed, that is its boundary is empty. (Not to be confused with the eventual boundaries which ##M## might have embedded in an ambient Euclidean space. A closed ball or sphere alone has no boundary. Think of the universe when people say it might be closed without a boundary.)
Then we have ##\int_M ν = \int_M dω \stackrel{Stokes}{=} \int_{\delta M} ω = \int_∅ ω = 0##.
I added this because it's the most elegant statement of Stokes' theorem which is about boundaries. Remember that you've been told you that Stokes' theorem and the chain rule are the most important facts in calculus!
You could say as well that a closed form is a cycle, a closed path. Going once around it and you will end up at the start. (But this is a rather stretched picture. However, I guess it is the origin of it.)

I used the term boundary operator for ##d## occasionally to show where the geometrical term 'closed' comes from.
 
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  • #6
fresh_42 said:
This comes from its topological, resp. geometrical origin. A closed differential form is a cycle ##C##, a exact differential form a boundary ##B## which establishes the short exact sequence: ##0 → B → C → C/B → 0##. (A exact form is always closed.) See also Wiki: "Loosely speaking, a cycle is a closed submanifold, a boundary is the boundary of a submanifold with boundary, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries." Geometrically it is related to simplicial complexes where the language comes from.

Edit: I omitted the filtration for simplicity.

Edit 2: The Graßmann algebra ##Λ V## is also called the exterior algebra of ##V##. It is the structure where exterior derivatives live. It is exterior because a inner product on ##V## would make it an algebra with a binary multiplication ##V \times V → V## whereas the exterior product is between the filtration of ##Λ V##, e.g. ##a ∧ b = (-1)^{ij} b ∧ a## for ##a \in Λ^i(V)## and ##b \in Λ^j(V)##.
How do you define the quotient C/B of forms?
 
  • #7
WWGD said:
How do you define the quotient C/B of forms?
As quotient of a Graßmann algebra and one of its ideals.
 

1. Why is the term "exterior" used in science?

The term "exterior" is used in science to describe something that is outside of a particular system or object. It is often used in contrast to the term "interior", which refers to something that is contained within the system or object.

2. What does "closed" mean in scientific terms?

In science, the term "closed" refers to a system or object that does not exchange matter or energy with its surroundings. This means that the total amount of matter and energy within the system remains constant.

3. Why is the term "exact" important in science?

The term "exact" is important in science because it implies a high level of precision and accuracy. When a measurement or calculation is described as "exact", it means that there is no margin of error and the value is known with complete certainty.

4. How are the terms "exterior", "closed", and "exact" related?

These terms are related because they all describe different aspects of a system or object. "Exterior" describes the location of something relative to a system, "closed" describes the exchange of matter and energy within a system, and "exact" describes the level of precision and accuracy of a measurement or calculation within a system.

5. What is the significance of these terms in scientific research?

These terms are important in scientific research because they help scientists to accurately describe and analyze systems and objects. By understanding the exterior, closed, and exact qualities of a system, scientists can make more accurate predictions and draw more reliable conclusions from their research.

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