1056: Derivation of Tensor Product of Tensors by Real Parameter Satisfies Leibniz Rule

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description/proof of that derivation of tensor product of tensors by real parameter satisfies Leibniz rule

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof

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Starting Context

• The reader knows a definition of tensor product of tensors.
• The reader knows a definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space.
• The reader knows a definition of convergence of map from topological space minus point into topological space with respect to point.
• The reader admits the proposition that for any map from any topological space minus any point into any finite-dimensional real vectors space with the canonical topology, the convergence of the map with respect to the point exists if and only if the convergences of the component maps (with respect to any basis) with respect to the point exist, and then, the convergence Is expressed with the convergences.
• The reader admits the proposition that for any field and any $k$ finite-dimensional vectors spaces over the field, the tensors space with respect to the field and the vectors spaces and the field has the basis that consists of the tensor products of the elements of the dual bases of any bases of the vectors spaces.

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Target Context

• The reader will have a description and a proof of the proposition that the derivation of the tensor product of any tensors by any real parameter satisfies the Leibniz rule.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T$: $=(r_1,r_2)\subseteq \mathbb{R}$, with the subspace topology with $\mathbb{R}$ as the Euclidean topological space
$r$: $\in T$
$\{V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}\}$: $\subseteq \{\text{ the finite-dimensional }\mathbb{R}\text{ vectors spaces }\}$
$L(V_{1,1},...,V_{1,k_1}:\mathbb{R})$: $=\text{ the tensors space }$, with the canonical topology
$L(V_{2,1},...,V_{2,k_2}:\mathbb{R})$: $=\text{ the tensors space }$, with the canonical topology
$t^1$: $:T\to L(V_{1,1},...,V_{1,k_1}:\mathbb{R})$
$t^2$: $:T\to L(V_{2,1},...,V_{2,k_2}:\mathbb{R})$
$L(V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}:\mathbb{R})$: $=\text{ the tensors space }$, with the canonical topology
$t^1\otimes t^2$: $:T\to L(V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}:\mathbb{R})$
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Statements:
$\mathrm{\exists }d{t}^1/dr\wedge \mathrm{\exists }d{t}^2/dr$
$⟹$
$\mathrm{\exists }d(t^1\otimes t^2)/dr\wedge d(t^1\otimes t^2)/dr=d{t}^1/dr\otimes t^2(r)+t^1(r)\otimes d{t}^2/dr$
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2: Proof

Whole Strategy: Step 1: for each of $\{V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}\}$, take any basis, $B_{j,l}=\{{b_{j,l}}_{m_{j,l}}\}$, and take the standard bases for $L(V_{1,1},...,V_{1,k_1}:\mathbb{R})$, $L(V_{2,1},...,V_{2,k_2}:\mathbb{R})$, and $L(V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}:\mathbb{R})$, $B_1$, $B_2$, and $B$; Step 2: let $t^1(r^{\prime })$, $t^2(r^{\prime })$, and $t^1(r^{\prime })\otimes t^2(r^{\prime })$ be expressed with the bases; Step 3: take $d(t^1\otimes t^2)/dr=li{m}_{r^{\prime }\to r}(t^1(r^{\prime })\otimes t^2(r^{\prime })-t^1(r)\otimes t^2(r))/(r^{\prime }-r)$ and apply the proposition that for any map from any topological space minus any point into any finite-dimensional real vectors space with the canonical topology, the convergence of the map with respect to the point exists if and only if the convergences of the component maps (with respect to any basis) with respect to the point exist, and then, the convergence Is expressed with the convergences.

Step 1:

For each of $\{V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}$, let us take any basis, $B_{j,l}=\{{b_{j,l}}_{m_{j,l}}\}$.

Let us take the standard basis for $L(V_{1,1},...,V_{1,k_1}:\mathbb{R})$, $B_1=\{b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\}$, which is possible by the proposition that for any field and any $k$ finite-dimensional vectors spaces over the field, the tensors space with respect to the field and the vectors spaces and the field has the basis that consists of the tensor products of the elements of the dual bases of any bases of the vectors spaces.

Let us take the standard basis for $L(V_{2,1},...,V_{2,k_2}:\mathbb{R})$, $B_2=\{b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}\}$, which is possible likewise.

Let us take the standard basis for $L(V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}:\mathbb{R})$, $B=\{b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}\}$, which is possible likewise.

Step 2:

$t^1(r^{\prime })=t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}$.

$t^2(r^{\prime })=t_{m_{2,1},...,m_{2,k_2}}^2(r^{\prime })b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}$.

So, $t^1(r^{\prime })\otimes t^2(r^{\prime })=(t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}})\otimes (t_{m_{2,1},...,m_{2,k_2}}^2(r^{\prime })b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}})$.

By the property of tensor product of tensors mentioned in Note for the definition of tensor product of tensors, $=t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })t_{m_{2,1},...,m_{2,k_2}}^2(r^{\prime })b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}$, which is the expansion of $t^1(r^{\prime })\otimes t^2(r^{\prime })$ with respect to $B$.

Step 3:

$d(t^1\otimes t^2)/dr=li{m}_{r^{\prime }\to r}(t^1(r^{\prime })\otimes t^2(r^{\prime })-t^1(r)\otimes t^2(r))/(r^{\prime }-r)$.

$=li{m}_{r^{\prime }\to r}(t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })t_{m_{2,1},...,m_{2,k_2}}^2(r^{\prime })b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}-t_{m_{1,1},...,m_{1,k_1}}^1(r)t_{m_{2,1},...,m_{2,k_2}}^2(r)b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}})/(r^{\prime }-r)=li{m}_{r^{\prime }\to r}(t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })t_{m_{2,1},...,m_{2,k_2}}^2(r^{\prime })-t_{m_{1,1},...,m_{1,k_1}}^1(r)t_{m_{2,1},...,m_{2,k_2}}^2(r))/(r^{\prime }-r)b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}$.

By the proposition that for any map from any topological space minus any point into any finite-dimensional real vectors space with the canonical topology, the convergence of the map with respect to the point exists if and only if the convergences of the component maps (with respect to any basis) with respect to the point exist, and then, the convergence Is expressed with the convergences, $d{t}_{m_{1,1},...,m_{1,k_1}}^1/dr=li{m}_{r^{\prime }\to r}(t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })-t_{m_{1,1},...,m_{1,k_1}}^1(r))/(r^{\prime }-r)$ and $d{t}_{m_{2,1},...,m_{2,k_2}}^2/dr=li{m}_{r^{\prime }\to r}(t_{m_{2,1},...,m_{2,k_2}}^2(r^{\prime })-t_{m_{2,1},...,m_{2,k_2}}^2(r))/(r^{\prime }-r)$ exist.

By the well-known fact in real analysis, $d(t_{m_{1,1},...,m_{1,k_1}}^1(r)t_{m_{2,1},...,m_{2,k_2}}^2(r))/dr=li{m}_{r^{\prime }\to r}(t_{m_{1,1},...,m_{1,k_1}}^1(r^{\prime })t_{m_{2,2},...,m_{2,k_2}}^2(r^{\prime })-t_{m_{1,1},...,m_{1,k_1}}^1(r)t_{m_{2,1},...,m_{2,k_2}}^2(r))/(r^{\prime }-r)$ exists and equals $d{t}_{m_{1,1},...,m_{1,k_1}}^1/dr{t}_{m_{2,1},...,m_{2,k_2}}^2(r)+t_{m_{1,1},...,m_{1,k_1}}^{1}d{t}_{m_{2,1},...,m_{2,k_2}}^2/dr$.

By the proposition that for any map from any topological space minus any point into any finite-dimensional real vectors space with the canonical topology, the convergence of the map with respect to the point exists if and only if the convergences of the component maps (with respect to any basis) with respect to the point exist, and then, the convergence Is expressed with the convergences, $d(t^1\otimes t^2)/dr=li{m}_{r^{\prime }\to r}(t^1(r^{\prime })\otimes t^2(r^{\prime })-t^1(r)\otimes t^2(r))/(r^{\prime }-r)$ exists and equals $(d{t}_{m_{1,1},...,m_{1,k_1}}^1/dr{t}_{m_{2,1},...,m_{2,k_2}}^2(r)+t_{m_{1,1},...,m_{1,k_1}}^1(r)d{t}_{m_{2,1},...,m_{2,k_2}}^2/dr)b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}$.

$=d{t}_{m_{1,1},...,m_{1,k_1}}^1/dr{t}_{m_{2,1},...,m_{2,k_2}}^2(r)b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}+t_{m_{1,1},...,m_{1,k_1}}^1(r)d{t}_{m_{2,1},...,m_{2,k_2}}^2/dr{b}_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}}\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}}=(d{t}_{m_{1,1},...,m_{1,k_1}}^1/dr{b}_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}})\otimes (t_{m_{2,1},...,m_{2,k_2}}^2(r)b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}})+(t_{m_{1,1},...,m_{1,k_1}}^1(r)b_{1,1}^{m_{1,1}}\otimes ...\otimes b_{1,k_1}^{m_{1,k_1}})\otimes (d{t}_{m_{2,1},...,m_{2,k_2}}^2/dr\otimes b_{2,1}^{m_{2,1}}\otimes ...\otimes b_{2,k_2}^{m_{2,k_2}})=d{t}^1/dr\otimes t^2(r)+t^1(r)\otimes d{t}^2/dr$.

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References

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