187 Smooth Assignment
A smooth mapping that attaches a tensor to each point that is used to define smooth tensor fields.
definition [d] (Smooth Assignment = Smooth Section) A mapping that attaches a type-\((r,s)\) tensor to each point so the assignment is smooth: for \(U \subset M\),
- \(T : U \rightarrow T^{r}_{\ s}(M)\) ,
- \(T(P) \equiv T_{P} \in T^{r}_{\ s,P}(M)\) ,
with component functions of class \(C^{\infty}\) in every chart.
where
- \(M\) is a smooth manifold.
- \(U\) is a subset of \(M\).
- \(T\) is the assignment.
- \(P\) is a point of \(M\).
- \(T^{r}_{\ s}(M)\) is the bundle of type-\((r,s)\) tensors on \(M\).
- \(T^{r}_{\ s,P}(M)\) is the space of type-\((r,s)\) tensors at \(P\).
- \(r\) is the contravariant degree.
- \(s\) is the covariant degree.
- \(C^{\infty}\) means infinitely differentiable.
Note:
- \(T\) is also called a tensor field.
- this is the definition of a smooth tensor field of type \((r,s)\).
definition [d] (Smooth Assignment = Cross Section = Section) A differentiable map \(s: M \rightarrow E\) into a vector bundle that lands in the fiber over each point:
- \(\pi \circ s = \mathrm{id}_{M}\) ,
so \(s(p) \in E_{p} = \pi^{-1}(p)\) for every \(p \in M\).
where
- \(M\) is the base manifold.
- \(E\) is the total space of the bundle.
- \(\pi: E \rightarrow M\) is the bundle projection.
- \(s\) is the section.
- \(\mathrm{id}_{M}\) is the identity map on \(M\).
- \(E_{p} = \pi^{-1}(p)\) is the fiber over \(p\).
- \(p\) is a point of \(M\).
Note:
- the bundle may be a tensor bundle.
- a smooth section of \(T^{r}_{\ s}(M)\) is precisely a smooth type-\((r,s)\) tensor field.
187.1 Elementary Example
187.1.1 Simple
A smooth assignment attaches a tensor value to each point of a finite base.
\[ U = \{ p,\ q,\ r \} \]
\[ T(p) = e_{1},\quad T(q) = e_{2},\quad T(r) = e_{1}+e_{2} \]
where
- \(T\) is the assignment.
- \(U\) is the set of points.
- \(e_{1}, e_{2}\) are values in a vector space at those points.
187.2 References
- Hassani, S. Mathematical Physics, 2nd ed. Springer. — tensor field as smooth assignment \(T:U\to T^{r}_{s}(M)\).
- Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — cross section \(s:M\to E\) with \(\pi\circ s=\mathrm{id}_{M}\).
- Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — smooth component fields.
- Alternating Function
- Alternating k-linear function
- Boundary Operation
- Bundle
- Cartesian Product
- Constant Metric Tensor
- Contravariant Rank
- Covariant Rank
- Covariant, Contravariant, and Coordinate Systems
- Differential Forms
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- Edge of a Domain
- Exterior Algebra and Exterior Derivatives
- Forms
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- Inverse Metric
- k-Covector
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- k-Fold Product
- k-form
- k-Tensor
- Lie
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- Lorentzian Manifold
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- Multilinear Function
- Nondegenerate
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- Smooth
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- Smooth Mapping
- Stoke’s Theorem and the Fundamental Theorem of Calculus
- Symmetric Array
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- Type-(0,2) Tensor Field
- Type-(q,r) Tensor