Различия

Показаны различия между двумя версиями страницы.

--- math:2-14 [2023/06/07 12:38] – root
+++ math:2-14 [2023/06/07 12:56] (текущий) – [47.7.] root
@@ Строка 5: / Строка 5: @@
 $F=e^{1}\otimes e_{2}+e^{2}\otimes (e_{1}+3e_3)\in\mathbb{T}_{1}^{1}\left(V\right),$
 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 $F$:
-$F(v,f)=e^{1}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes e_{2}\left(e^{1}+e^{2}+e^{3}\right)+e^{2}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes (e_{1}\left(e^{1}+e^{2}+e^{3}\right)+3e_{3}\left(e^{1}+e^{2}+e^{3}\right))=$
+$$F(v,f)=e^{1}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes e_{2}\left(e^{1}+e^{2}+e^{3}\right)+e^{2}\left(e_{1}+5e_{2}+4e_{3}\right)\otimes (e_{1}\left(e^{1}+e^{2}+e^{3}\right)+3e_{3}\left(e^{1}+e^{2}+e^{3}\right))=$$
-$e^{1}\left(e_{1}\right)\otimes e_{2}\left(e^{2}\right)+e^{2}\left(5e_{2}\right)\otimes (e_{1}\left(e^{1}\right)+3e_{3}\left(e^{3}\right))=1\otimes 1+5\otimes 4=21$
+$$e^{1}\left(e_{1}\right)\otimes e_{2}\left(e^{2}\right)+e^{2}\left(5e_{2}\right)\otimes (e_{1}\left(e^{1}\right)+3e_{3}\left(e^{3}\right))=1\otimes 1+5\otimes 4=21.$$
 ==== 47.3. ====
@@ Строка 19: / Строка 21: @@
 === a) ===
-$A=e_{1}\otimes e2+e2\otimes e3+e2\otimes e2$
+$A=e_{1}\otimes e_2+e_2\otimes e_3+e_2\otimes e_2$
-$B=e1\otimes e2\otimes \left(e1-e3\right)$
+$B=e_1\otimes e_2\otimes \left(e_1-e_3\right)$
-$v_{1}=e1,v2=e1+e2,v3=e2+e3,v4=v5=e2$
+$v_{1}=e_1,v_2=e_1+e_2,v_3=e_2+e_3,v_4=v_5=e_2$
-$B=e1\otimes e2\otimes \left(e1-e3\right)$
+-----
-$A=e1\left(e1\right)\otimes e2\left(e1+e2\right)+e2\left(e1\right)\otimes e3\left(e1+e2\right)+e2\left(e1\right)\otimes e2\left(e1+e2\right)=$
+$B=e_1\otimes e_2\otimes \left(e_1-e_3\right)$
+$A=e_1\left(e_1\right)\otimes e_2\left(e_1+e_2\right)+e_2\left(e_1\right)\otimes e_3\left(e_1+e_2\right)+e_2\left(e_1\right)\otimes e_2\left(e_1+e_2\right)=$
 $1\otimes 1+0\otimes 0+0\otimes 1=1$
-$B=e1\left(e2+e3\right)\otimes e2\left(e2\right)\otimes \left(e1\left(e2\right)-e3\left(e2\right)\right)=0\otimes 1\otimes 0=0$
+$B=e_1\left(e_2+e_3\right)\otimes e_2\left(e_2\right)\otimes \left(e_1\left(e_2\right)-e_3\left(e_2\right)\right)=0\otimes 1\otimes 0=0$
 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 45: / Строка 49: @@
 === б) ===
-$A=e1\otimes e2+e2\otimes e3+e3\otimes e1$
+$A=e_1\otimes e_2+e_2\otimes e_3+e_3\otimes e_1$
+$B=\sum e_i\otimes e_j\otimes e_k$
-$B=\sum ei\otimes ej\otimes ek$
+$v_1=e_1+e_2,v_2=e_3+e_2,v_3=e_1+e_3,v_4=v_5=e_2$
-$v1=e1+e2,v2=e3+e2,v3=e1+e3,v4=v5=e2$
+-----
-$A=e1\left(e1+e2\right)\otimes e2\left(e3+e2\right)+e2\left(e1+e2\right)\otimes e3\left(e3+e2\right)+e3\left(e1+e2\right)\otimes e1\left(e3+e2\right)=$
+$A=e_1\left(e_1+e_2\right)\otimes e_2\left(e_3+e_2\right)+e_2\left(e_1+e_2\right)\otimes e_3\left(e_3+e_2\right)+e_3\left(e_1+e_2\right)\otimes e1\left(e_3+e_2\right)=$
 $1\otimes 1+1\otimes 1+0\otimes 0=2$
-$B=\sum ei\otimes ej\otimes ek=ei\left(e1+e3\right)\otimes ej\left(e2\right)\otimes ek\left(e2\right)=2$
+$B=\sum e_i\otimes e_j\otimes e_k=e_i\left(e_1+e_3\right)\otimes e_j\left(e_2\right)\otimes e_k\left(e_2\right)=2$
-$A=e1\left(e2\right)\otimes e2\left(e2\right)+e2\left(e2\right)\otimes e3\left(e2\right)+e3\left(e2\right)\otimes e1\left(e2\right)=0\otimes 1+1\otimes 0+0\otimes 0=0$
+$A=e_1\left(e_2\right)\otimes e_2\left(e_2\right)+e_2\left(e_2\right)\otimes e_3\left(e_2\right)+e_3\left(e_2\right)\otimes e_1\left(e_2\right)=0\otimes 1+1\otimes 0+0\otimes 0=0$
-$B=\sum ei\otimes ej\otimes ek=ei\left(e1+e2\right)\otimes ej\left(e3+e2\right)\otimes ek\left(e1+e3\right)=8$
+$B=\sum e_i\otimes e_j\otimes e_k=e_i\left(e_1+e_2\right)\otimes e_j\left(e_3+e_2\right)\otimes e_k\left(e_1+e_3\right)=8$
 Тогда
@@ Строка 80: / Строка 86: @@
 и $B\otimes A$ тензоров
-$A=e1\otimes e2+e3\otimes e3$
+$A=e_1\otimes e_2+e_3\otimes e_3$
-$B=\det\left(v1,v2,v3\right)$
+$B=\det\left(v_1,v_2,v_3\right)$
-$A\otimes B\left(e1,e2,e3,e3,e3\right)=A\left(e1,e2\right)\otimes B\left(e3,e3,e3\right)=A\left(e1,e2\right)\otimes 0=0$
+$A\otimes B\left(e_1,e_2,e_3,e_3,e_3\right)=A\left(e_1,e_2\right)\otimes B\left(e_3,e_3,e_3\right)=A\left(e_1,e_2\right)\otimes 0=0$
-$B\otimes A\left(e1,e2,e3,e3,e3\right)=B\left(e1,e2,e3\right)\otimes A\left(e3,e3\right)=B\left(e1,e2,e3\right)\otimes \left(e1\left(e3\right)\otimes e2\left(e3\right)+e3\left(e3\right)\otimes e3\left(e3\right)\right)=1\otimes (0+1)=1$
+$B\otimes A\left(e_1,e_2,e_3,e_3,e_3\right)=B\left(e_1,e_2,e_3\right)\otimes A\left(e_3,e_3\right)=B\left(e_1,e_2,e_3\right)\otimes \left(e_1\left(e_3\right)\otimes e_2\left(e_3\right)+e_3\left(e_3\right)\otimes e_3\left(e_3\right)\right)=1\otimes (0+1)=1$
 ==== 47.7. ====
@@ Строка 96: / Строка 102: @@
 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}
+$$(e_{1},e_{2})=(\widetilde{e_{1}},\widetilde{e_{2}})\left(\begin{array}{cc}
 & 1\\
 & 3
@@ Строка 107: / Строка 113: @@
 & -1\\
 -2 & 1
-\end{array}\right)$
+\end{array}\right)$$
-$(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:
+==== 47.13. ====
+Найти полную свертку тензоров:
 === a) ===