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Alex M.
Alex M.
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Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $\omega$ along $c$ as

$$I(\omega, c) = \int _0 ^1 \omega_{c(t)} (\dot c (t)) \ \mathrm d t \ .$$

This is clear, but this is just a formula that does not give any insight into the innards of the concept.

Is is possible to define the concept of line integral by purely abstract properties?

To give two analogies, the algebraic tensor product is defined by some universal property, and then it is shown that it exists and is essentially unique. Similarly, the Haar measure on locally-compact groups is defined as a regular, positive measure, that is invariant under (left) translations, and then is shown to exist and be essentially unique. Do you know of any similar approach for line integrals?

To clarify: if $\mathcal C$ is the space of smooth curves in $M$, I am attempting to understand line integration as a map $I : \Omega ^1 (M) \times \mathcal C \to \mathbb R$ uniquely determined by some properties: what are these properties? For sure, linearity in the first argument is among them. What else is needed?

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $\omega$ along $c$ as

$$I(\omega, c) = \int _0 ^1 \omega_{c(t)} (\dot c (t)) \ \mathrm d t \ .$$

This is clear, but this is just a formula that does not give any insight into the innards of the concept.

Is is possible to define the concept of line integral by purely abstract properties?

To give two analogies, the algebraic tensor product is defined by some universal property, and then it is shown that it exists and is essentially unique. Similarly, the Haar measure on locally-compact groups is defined as a regular, positive measure, that is invariant under (left) translations, and then is shown to exist and be essentially unique. Do you know of any similar approach for line integrals?

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $\omega$ along $c$ as

$$I(\omega, c) = \int _0 ^1 \omega_{c(t)} (\dot c (t)) \ \mathrm d t \ .$$

This is clear, but this is just a formula that does not give any insight into the innards of the concept.

Is is possible to define the concept of line integral by purely abstract properties?

To give two analogies, the algebraic tensor product is defined by some universal property, and then it is shown that it exists and is essentially unique. Similarly, the Haar measure on locally-compact groups is defined as a regular, positive measure, that is invariant under (left) translations, and then is shown to exist and be essentially unique. Do you know of any similar approach for line integrals?

To clarify: if $\mathcal C$ is the space of smooth curves in $M$, I am attempting to understand line integration as a map $I : \Omega ^1 (M) \times \mathcal C \to \mathbb R$ uniquely determined by some properties: what are these properties? For sure, linearity in the first argument is among them. What else is needed?

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Alex M.
Alex M.
  • 5.4k
  • 11
  • 35
  • 52

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $\omega$ along $c$ as

$$I(\omega, c) = \int _0 ^1 \omega_{c(t)} (\dot c (t)) \ \mathrm d t \ .$$

This is clear, but this is just a formula that does not give any insight into the innards of the concept.

Is is possible to define the concept of line integral by purely abstract properties?

To give two analogies, the algebraic tensor product is defined by some universal property, and then it is shown that it exists and is essentially unique. Similarly, the Haar measure on locally-compact groups is defined as a regular, positive measure, that is invariant under (left) translations, and then is shown to exist and be essentially unique. Do you know of any similar approach for line integrals?