Transcription of Vector Algebra and Calculus - phys.ufl.edu
1 Vector Algebra and CalculusComponents:A=Ax i+Ay j+Az k = x i+ y j+ z kScalar ( Dot ) Product:A B=AxBx+AyBy+AzBz A= Ax x+ Ay y+ Az zVector ( Cross ) Product:A B=(AyBz AzBy) i+ (AzBx AxBz) j+ (AxBy AyBx) k A= Az y Ay z i+ Ax z Az x j+ Ay x Ax y kSome Identities:A (B C) = (A B) C)A (B C) = (A C)B (A B)CIt is sometimes convenient to represent the component labels of a Vector by asubscript which takes the values 1,2,3 for componentsx, y, z. Then we can writethe scalar product, for example, asA B=3Xi= Convention: Repeated indices are automaticallysummed. ThusA B= Delta: ij= 1 fori=j0 fori6= Epsilon Symbol: Let ijk= +1 (ijk) an even permutation of (123) 1 (ijk) an odd permutation of (123)0 two or more ofi, j, kare (A B)i= : ijk mnk= im jn in : When applying Vector identities to expressions involving the gradientdifferential operator care must be taken that the derivatives act on all functionsto their right.
2 For example: (B C) = xi(BCi) xi(BiC)Gauss Theorem:ZdV E=IdA EStokes Theorem:ZdA E=Idl EPotential Difference:P2ZP1dl = (P2) (P1)2
