XM are coordinates 4 Scalar functionson spacetimeµ O l 2,3not components of a crater

Uecter function number on function

Um m U Undofy oxm

ComponentsT basisvectors

Codector vectors number or function

W WudxnComponents I basiscooeaters

If we did then Wcv UCO

dxn 04 7 44M ofWW Wendy w v WµVdI WnwDX v V W lov Wu

Tensors

U Un n is a type f tensorcan be acted upon by 1 co vector

W weedXM is a type 9 tensorcan act on 1 vector

type nm teaser can be acteduponby n covectorscan act on in vectors

L teaser T The dead DX nW Tcu v D Trap dx4vddxkv.tw 9ox4

T1pVYufWuwlTIwlTnapdxaxOdxPxo0fyr WntMarsdxa dxBTl V ftp.dxaxodxfxoglyn VT

TrapUP dxd u TL

T H m LThem N

tcml LTIpdxaxodye.IO KMrP9xox09xd

TIaMTfIdfofTIpMdP xuTIoMK µ

not the same

T Ll N ilTrap Nfu components ofacovector

TINI NCT's

9µV metric 9 tenser

g inverse metric f tense

gmgv e.sicomponentsof f tensor

deil

identity tensor

gµTap Trap gudTL Tmpgrug

g

9 d't Sure

n Ed

SiuMM Mmm traceof Mfsindex contraction

T l'drap The

tension1nap MM UM

µ free index takes on 0,1 2 OrB

dip repeated indices summed overfree indices must balance same ineach term

repeated indices eachterm in pairs oneup onedown

Tap MH vieTap MM V

V

TI Mors w

f sane

lower the index

gavTureMH 9mVnewTarp nor Vu TurpsM

TrigppMM VuTur P M p Vu

no M p Un

Ture Me Vu

Mf MerRiemann tenser b

Rivals

Covariant Derivative On nm nm

PnfTLpMH fµV

ftp 1MH TLPPMH qvrPutman MMtT PnMM PnV04Th MH pmVV

PMtua mm Pmf

go.OMT.apmdtgarlmwtg

ofoncg.ws

gas o P gov

0µg oµ4TrapM4 on uT ddp to

PrPu torta in general

for 0 0 0qq.V Ppbvn Rnvs.pt0a0pWuPpPaWrRnvapWTLevi Civita tensor

Identity tensor of gM

Tensor f MinsMtv

V.V

original condo xn new cords In

Iii to IEIIi kEIiosIi.nIT iiJY qx of FINI oxI

ox ON JB 9 2 gyu gyu

JT l n nachoLevi Civita tensor type 9 tenserCmap

FT if Muap mean even oddpermutationof oil 2,3

0 otherwise

totally antisymmetric Enya EmptEarpEµpva

Emma µ 9i9 h exam

EEE i m iE nfmh nE m FE

EMMYMy pgtdet x if Matheny is

an evenherddpermof0 I 2

o otherwise

In.m h m'f m9aag det TgFF detoxify FJ

My µ µIFJ it MMM344 is beyond

Permof 01112,3o otherwise

Levi Civita tenser