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1、附A張量分析§A-1指標符號例如,三維空間任意一點P在笛卡兒坐標系x,x,x123用指標符x,i?1,2,3號表示為i數(shù)a,a,a,???,aa,i?1,2,???,n123nix,x,x,???,xx,i?1,2,???,n123ni變量指標符號i—指標——取值范圍為小于或等于n的所有正整數(shù)n—維數(shù)§A-1指標符號A張量分析一、求和約定和啞指標S?ax?ax????ax1122nnnnS??ax??ax求和指標iijj與所用的i?1j?1字母無關(guān)指標重復約定S?ax?ax只能一次iijj指標范圍用拉丁字母表示3維，希臘字母表2維§A-1指標符號一、求和約定和啞指標33??Ax
2、y雙重求和ijiji?1j?1Axy?Axy?Axy?Axy?ijij111112121313Axy?Axy?Axy?212122222323Axy?Axy?Axy313132323333Axyz代表27項ijkijk的和式§A-1指標符號二、自由指標Ax?Ax?Ax?b1111221331Ax?Ax?Ax?b2112222332Ax?Ax?Ax?b3113223333筒寫為Ax?bijjij——啞指標i——自由指標，在每一項中只出現(xiàn)一次，一個公式中必須相同§A-1指標符號三、Kronecker-?符號和置換符號(Ricci符號)Kronecker-?符號定義?1當i?j?ji
3、??ij???0當i?j當i,j?1,2,3時，有??????1112233????????????0122123323113§A-1指標符號三、Kronecker-?符號和置換符號(Ricci符號)Kronecker-?符號定義???100111213??????010?1ij212223???001313233?a??a??a??a?aijji11i22i33i?A?Aimmjij????????3ii112233????ikkjij???????3ijijiijj?????ijjkklila??aikkjija??a?a?a?aijijii112233a??aiijj直角
4、坐標系的e?e??ijij基矢量§A-1指標符號三、Kronecker-?符號和置換符號(Ricci符號)Ricci符號定義偶次置換?1若?i,j,k???1,2,3?,?2,3,1?,?3,1,2??eijk???1若?i,j,k???3,2,1?,?2,1,3?,?1,3,2???0若有兩個或三個指標相等奇次置換e?e?e?1123231312e?e?e??1213132321e?e?e?????0111112113§A-1指標符號三、Kronecker-?符號和置換符號(Ricci符號)Ricci符號定義??????i1i2i3i1j1k1e????????ijkj1j2
5、j3i2j2k2??????k1k2k3i3j3k3???001313233e?????010??1321212223???100111213e??e??e??eijkjikikjkjie?e?eijkjkikij??????i1i2i3p1q1r1ee????????ijkpqrj1j2j3p2q2r2??????k1k2k3p3q3r3?????????????i1p1i2p2i3p3i1p1ip???ipiqiree????ijkpqrjpjqjr???kpkqkrp?k??iqiree???????ijkkqriqjrirjq??jqjraaa111213A?aaa?a
6、aa?aaa?212223112233122331aaa313233aaa?aaa?aaa?aaa132132132231122133112332?eaaa?eaaaijk1i2j3kijki1j2k3Kronecker-?和Ricci符號的關(guān)系ee??????kijkstisjtjsitA張量分析§A-2矢量的基本運算在三維空間中,任意矢量都可以表示為三個基e,e,e123矢量的線性組合a?ae?ae?ae?ae112233iiai為矢量a在基矢量ei下的分解系數(shù),也稱矢量的分量一、矢量點積e?e??ijijA張量分析§A-2矢量的基本運算一、矢量點積a?b?ae?be?ab
7、?iijjijijab?abiijj二、矢量叉積e?e?eeijijkk§A-2矢量的基本運算A張量分析二、矢量叉積e??eiikk證明e??ejjkk???i1i2i3e?e????ijj1j2j3eee123?e??e?ee?eerstirjstijttijkk§A-2矢量的基本運算A張量分析二、矢量叉積a?b?ae?be?abe?eiijjijijabee?eabe?cijijkkijkijkc?eabkijkij§A-2矢量的基本運算A張量分析三、矢量的混合積a?b?c?eabe?c
