Различия

Показаны различия между двумя версиями страницы.

--- math:2-14 [2023/06/07 12:36] – создано root
+++ math:2-14 [2023/06/07 12:56] (текущий) – [47.7.] root
@@ Строка 1: / Строка 1: @@
-.2. Найти значение $F(v,f)$ тензора
+==== 47.2. ====
-$F=e^{1}\otimes e_{2}+e^{2}\otimes (e_{1}+\lyxmathsym{З}\lyxmathsym{е}_{3})\in\mathbb{T}_{1}^{1}\left(V\right),$
+Найти значение $F(v,f)$ тензора
+$F=e^{1}\otimes e_{2}+e^{2}\otimes (e_{1}+3e_3)\in\mathbb{T}_{1}^{1}\left(V\right),$
 где $v=e_{1}+5e_{2}+4e_{3}$, $f=e^{1}+e^{2}+e^{3}$.
+-----
 Для того чтобы вычислить $F(v,f)$, нужно подставить значения $v$
 и $f$ в $F$:
-$F(v,f)=e^{1}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes e_{2}\left(e^{1}+e^{2}+e^{3}\right)+e^{2}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes (e_{1}\left(e^{1}+e^{2}+e^{3}\right)+3e_{3}\left(e^{1}+e^{2}+e^{3}\right))=$
+$$F(v,f)=e^{1}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes e_{2}\left(e^{1}+e^{2}+e^{3}\right)+e^{2}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes (e_{1}\left(e^{1}+e^{2}+e^{3}\right)+3e_{3}\left(e^{1}+e^{2}+e^{3}\right))=$$
-$e^{1}\left(e_{1}\right)\otimes e_{2}\left(e^{2}\right)+e^{2}\left(5e_{2}\right)\otimes (e_{1}\left(e^{1}\right)+3e_{3}\left(e^{3}\right))=1\otimes 1+5\otimes 4=21$
+$$e^{1}\left(e_{1}\right)\otimes e_{2}\left(e^{2}\right)+e^{2}\left(5e_{2}\right)\otimes (e_{1}\left(e^{1}\right)+3e_{3}\left(e^{3}\right))=1\otimes 1+5\otimes 4=21.$$
-.3. Найти значение тензора $A\otimes B-B\otimes A\in\mathbb{T}_{5}^{0}\left(V\right)$
+==== 47.3. ====
+Найти значение тензора $A\otimes B-B\otimes A\in\mathbb{T}_{5}^{0}\left(V\right)$
 от набора
-a) $A=e_{1}\otimes e2+e2\otimes e3+e2\otimes e2$
+=== a) ===
-$B=e1\otimes e2\otimes \left(e1-e3\right)$
+$A=e_{1}\otimes e_2+e_2\otimes e_3+e_2\otimes e_2$
-$v_{1}=e1,v2=e1+e2,v3=e2+e3,v4=v5=e2$
+$B=e_1\otimes e_2\otimes \left(e_1-e_3\right)$
-$B=e1\otimes e2\otimes \left(e1-e3\right)$
+$v_{1}=e_1,v_2=e_1+e_2,v_3=e_2+e_3,v_4=v_5=e_2$
-$A=e1\left(e1\right)\otimes e2\left(e1+e2\right)+e2\left(e1\right)\otimes e3\left(e1+e2\right)+e2\left(e1\right)\otimes e2\left(e1+e2\right)=$
+-----
+$B=e_1\otimes e_2\otimes \left(e_1-e_3\right)$
+$A=e_1\left(e_1\right)\otimes e_2\left(e_1+e_2\right)+e_2\left(e_1\right)\otimes e_3\left(e_1+e_2\right)+e_2\left(e_1\right)\otimes e_2\left(e_1+e_2\right)=$
 $1\otimes 1+0\otimes 0+0\otimes 1=1$
-$B=e1\left(e2+e3\right)\otimes e2\left(e2\right)\otimes \left(e1\left(e2\right)-e3\left(e2\right)\right)=0\otimes 1\otimes 0=0$
+$B=e_1\left(e_2+e_3\right)\otimes e_2\left(e_2\right)\otimes \left(e_1\left(e_2\right)-e_3\left(e_2\right)\right)=0\otimes 1\otimes 0=0$
 тогда $A\otimes B(v)=0$
-$B=e1\left(e1\right)\otimes e2\left(e1+e2\right)\otimes \left(e1\left(e2+e3\right)-e3\left(e2+e3\right)\right)=1\otimes 1\otimes \left(-1\right)=-1$
+$B=e_1\left(e_1\right)\otimes e_2\left(e_1+e_2\right)\otimes \left(e_1\left(e_2+e_3\right)-e_3\left(e_2+e_3\right)\right)=1\otimes 1\otimes \left(-1\right)=-1$
-$A=e1\left(e2\right)\otimes e2\left(e2\right)+e2\left(e2\right)\otimes e3\left(e2\right)+e2\left(e2\right)\otimes e2\left(e2\right)=0\otimes 1+1\otimes 0+1\otimes 1=1$
+$A=e_1\left(e_2\right)\otimes e_2\left(e_2\right)+e_2\left(e_2\right)\otimes e_3\left(e_2\right)+e_2\left(e_2\right)\otimes e_2\left(e_2\right)=0\otimes 1+1\otimes 0+1\otimes 1=1$
 тогда $B\otimes A(v)=-1$
@@ Строка 38: / Строка 47: @@
 следовательно $\left(A\otimes B-B\otimes A\right)\left(v\right)=0-(-1)=1$
-б) $A=e1\otimes e2+e2\otimes e3+e3\otimes e1$
+=== б) ===
-$B=\sum ei\otimes ej\otimes ek$
+$A=e_1\otimes e_2+e_2\otimes e_3+e_3\otimes e_1$
-$v1=e1+e2,v2=e3+e2,v3=e1+e3,v4=v5=e2$
+$B=\sum e_i\otimes e_j\otimes e_k$
-$A=e1\left(e1+e2\right)\otimes e2\left(e3+e2\right)+e2\left(e1+e2\right)\otimes e3\left(e3+e2\right)+e3\left(e1+e2\right)\otimes e1\left(e3+e2\right)=$
+$v_1=e_1+e_2,v_2=e_3+e_2,v_3=e_1+e_3,v_4=v_5=e_2$
+-----
+$A=e_1\left(e_1+e_2\right)\otimes e_2\left(e_3+e_2\right)+e_2\left(e_1+e_2\right)\otimes e_3\left(e_3+e_2\right)+e_3\left(e_1+e_2\right)\otimes e1\left(e_3+e_2\right)=$
 $1\otimes 1+1\otimes 1+0\otimes 0=2$
-$B=\sum ei\otimes ej\otimes ek=ei\left(e1+e3\right)\otimes ej\left(e2\right)\otimes ek\left(e2\right)=2$
+$B=\sum e_i\otimes e_j\otimes e_k=e_i\left(e_1+e_3\right)\otimes e_j\left(e_2\right)\otimes e_k\left(e_2\right)=2$
-$A=e1\left(e2\right)\otimes e2\left(e2\right)+e2\left(e2\right)\otimes e3\left(e2\right)+e3\left(e2\right)\otimes e1\left(e2\right)=0\otimes 1+1\otimes 0+0\otimes 0=0$
+$A=e_1\left(e_2\right)\otimes e_2\left(e_2\right)+e_2\left(e_2\right)\otimes e_3\left(e_2\right)+e_3\left(e_2\right)\otimes e_1\left(e_2\right)=0\otimes 1+1\otimes 0+0\otimes 0=0$
-$B=\sum ei\otimes ej\otimes ek=ei\left(e1+e2\right)\otimes ej\left(e3+e2\right)\otimes ek\left(e1+e3\right)=8$
+$B=\sum e_i\otimes e_j\otimes e_k=e_i\left(e_1+e_2\right)\otimes e_j\left(e_3+e_2\right)\otimes e_k\left(e_1+e_3\right)=8$
 Тогда
@@ Строка 58: / Строка 71: @@
 $A\otimes B-B\otimes A=2\otimes 2-8\otimes 0=4$
-.4. Найти значение $F\left(v,v,v,f,f\right)$ тензора $F\in\mathbb{T}_{3}^{2}\left(V\right)$,
+==== 47.4. ====
+Найти значение $F\left(v,v,v,f,f\right)$ тензора $F\in\mathbb{T}_{3}^{2}\left(V\right)$,
 если все координаты тензора $F$ равны 3, и
@@ Строка 67: / Строка 82: @@
 $F=3\sum e^{i}\left(e_{1}+2e_{2}+3e_{3}+4e_{4}\right)\otimes e^{j}\left(e_{1}+2e_{2}+3e_{3}+4e_{4}\right)\otimes e^{k}\left(e_{1}+2e_{2}+3e_{3}+4e_{4}\right)\otimes e_{l}\left(e^{1}-e^{4}\right)\otimes e_{m}\left(e^{1}-e^{4}\right)=0$
-.6. Найти координаты с индексами 1, 2, 3, 3, 3 произведений $A\otimes B$
+==== 47.6. ====
+Найти координаты с индексами 1, 2, 3, 3, 3 произведений $A\otimes B$
 и $B\otimes A$ тензоров
-$A=e1\otimes e2+e3\otimes e3$
+$A=e_1\otimes e_2+e_3\otimes e_3$
-$B=\det\left(v1,v2,v3\right)$
+$B=\det\left(v_1,v_2,v_3\right)$
-$A\otimes B\left(e1,e2,e3,e3,e3\right)=A\left(e1,e2\right)\otimes B\left(e3,e3,e3\right)=A\left(e1,e2\right)\otimes 0=0$
+$A\otimes B\left(e_1,e_2,e_3,e_3,e_3\right)=A\left(e_1,e_2\right)\otimes B\left(e_3,e_3,e_3\right)=A\left(e_1,e_2\right)\otimes 0=0$
-$B\otimes A\left(e1,e2,e3,e3,e3\right)=B\left(e1,e2,e3\right)\otimes A\left(e3,e3\right)=B\left(e1,e2,e3\right)\otimes \left(e1\left(e3\right)\otimes e2\left(e3\right)+e3\left(e3\right)\otimes e3\left(e3\right)\right)=1\otimes (0+1)=1$
+$B\otimes A\left(e_1,e_2,e_3,e_3,e_3\right)=B\left(e_1,e_2,e_3\right)\otimes A\left(e_3,e_3\right)=B\left(e_1,e_2,e_3\right)\otimes \left(e_1\left(e_3\right)\otimes e_2\left(e_3\right)+e_3\left(e_3\right)\otimes e_3\left(e_3\right)\right)=1\otimes (0+1)=1$
-.7. Найти координаты:
+==== 47.7. ====
+Найти координаты:
-a) $\widetilde{t_{21}^{1}}$ тензора $e^{1}\otimes e^{2}\otimes (e_{1}+e_{2})\in T_{2}^{1}(V)$
+=== a) ===
+$\widetilde{t_{21}^{1}}$ тензора $e^{1}\otimes e^{2}\otimes (e_{1}+e_{2})\in T_{2}^{1}(V)$
 в базисе
-$(\widetilde{e_{1}},\widetilde{e_{2}})=(e_{1},e_{2})\left(\begin{array}{cc}
+$$(\widetilde{e_{1}},\widetilde{e_{2}})=(e_{1},e_{2})\left(\begin{array}{cc}
 & 1\\
 & 3
-\end{array}\right)$
+\end{array}\right)$$
-$(e_{1},e_{2})=(\widetilde{e_{1}},\widetilde{e_{2}})\left(\begin{array}{cc}
+$$(e_{1},e_{2})=(\widetilde{e_{1}},\widetilde{e_{2}})\left(\begin{array}{cc}
 & 1\\
 & 3
@@ Строка 94: / Строка 113: @@
 & -1\\
 -2 & 1
-\end{array}\right)$
+\end{array}\right)$$
-$(e^{1},e^{2})=(\widetilde{e^{1}},\widetilde{e^{2}})\left(\begin{array}{cc}
+$$(e^{1},e^{2})=(\widetilde{e^{1}},\widetilde{e^{2}})\left(\begin{array}{cc}
 & 1\\
 & 3
-\end{array}\right)$
+\end{array}\right)$$
+$$\widetilde{t_{jk}^{i}}=\left(\widetilde{e^{1}}+2\widetilde{e^{2}}\right)\otimes \left(\widetilde{e^{1}}+3\widetilde{e^{2}}\right)\otimes (3\widetilde{e_{1}}-2\widetilde{e_{2}}-\widetilde{e_{1}}+\widetilde{e_{2}})=\left(\widetilde{e^{1}}+2\widetilde{e^{2}}\right)\otimes \left(\widetilde{e^{1}}+3\widetilde{e^{2}}\right)\otimes (2\widetilde{e_{1}}-\widetilde{e_{2}})$$
-$\widetilde{t_{jk}^{i}}=\left(\widetilde{e^{1}}+2\widetilde{e^{2}}\right)\otimes \left(\widetilde{e^{1}}+3\widetilde{e^{2}}\right)\otimes (3\widetilde{e_{1}}-2\widetilde{e_{2}}-\widetilde{e_{1}}+\widetilde{e_{2}})=\left(\widetilde{e^{1}}+2\widetilde{e^{2}}\right)\otimes \left(\widetilde{e^{1}}+3\widetilde{e^{2}}\right)\otimes (2\widetilde{e_{1}}-\widetilde{e_{2}})$
+$$\widetilde{t_{21}^{1}}=\left(\widetilde{e^{1}}+2\widetilde{e^{2}}\right)\otimes \left(\widetilde{e^{1}}+3\widetilde{e^{2}}\right)\otimes (2\widetilde{e_{1}}-\widetilde{e_{2}})\left(\widetilde{e_{2}},\widetilde{e_{1}},\widetilde{e^{1}}\right)=$$
-$\widetilde{t_{21}^{1}}=\left(\widetilde{e^{1}}+2\widetilde{e^{2}}\right)\otimes \left(\widetilde{e^{1}}+3\widetilde{e^{2}}\right)\otimes (2\widetilde{e_{1}}-\widetilde{e_{2}})\left(\widetilde{e_{2}},\widetilde{e_{1}},\widetilde{e^{1}}\right)=$
+$$\left(\widetilde{e^{1}}\left(\widetilde{e_{2}}\right)+2\widetilde{e^{2}}\left(\widetilde{e_{2}}\right)\right)\otimes \left(\widetilde{e^{1}}\left(\widetilde{e_{1}}\right)+3\widetilde{e^{2}}\left(\widetilde{e_{1}}\right)\right)\otimes (2\widetilde{e_{1}}\left(\widetilde{e^{1}}\right)-\widetilde{e_{2}}\left(\widetilde{e^{1}}\right))=$$
-$\left(\widetilde{e^{1}}\left(\widetilde{e_{2}}\right)+2\widetilde{e^{2}}\left(\widetilde{e_{2}}\right)\right)\otimes \left(\widetilde{e^{1}}\left(\widetilde{e_{1}}\right)+3\widetilde{e^{2}}\left(\widetilde{e_{1}}\right)\right)\otimes (2\widetilde{e_{1}}\left(\widetilde{e^{1}}\right)-\widetilde{e_{2}}\left(\widetilde{e^{1}}\right))=$
+$$\left(0+2\right)\otimes \left(1+3\cdot0\right)\otimes (2-0)=2\otimes 1\otimes 2=4$$
-$\left(0+2\right)\otimes \left(1+3\cdot0\right)\otimes (2-0)=2\otimes 1\otimes 2=4$
+==== 47.13. ====
+Найти полную свертку тензоров:
-.13. Найти полную свертку тензоров:
+=== a) ===
-a) $t_{i}^{j}=(e_{1}+3e_{2}-e_{3})\otimes (e^{1}-2e^{3}+3e^{4})-(e_{1}+e_{3})\otimes (e^{1}-3e^{3}+e^{4})$;
+$t_{i}^{j}=(e_{1}+3e_{2}-e_{3})\otimes (e^{1}-2e^{3}+3e^{4})-(e_{1}+e_{3})\otimes (e^{1}-3e^{3}+e^{4})$;
 $t_{i}^{i}=(e_{1})\otimes (e^{1})+(-e_{3})\otimes (-2e^{3})-(e_{1})\otimes (e^{1})-(e_{3})\otimes (-3e^{3})=1+2-1+3=5$