81orn.ec'r' OH
, .. , • A - ':,pm oH
""' !, " " . "T 40
c1: P s RSno\'.>i,..o (o
13 Nc:,v 2023
• G ca.tors - rna gnih.Jde, s Peed
L-, h.oro-r"i bY""I '. IX , (3 / Y
vect:-oy- - or,-en-to-t-i o.- ~,..,.~ mag.--..·1--\--\..<le L;"'-e
L, u
.......,..__ I ~ ., U (r---r~"
UJ~ ,J
tensor ·. o Y'T"agn...i +vde \Ni\:-n !WO di-rections
~ .6., ~, A C~ca~)
t e.GY"\-0,i,c.o.,~ -e-ve~n·, ~ ·,s o. -+-e~ OY-
ot-"'- ~ ~r:g.e.. -bensor - ? Scalar-
st
I deq~ te.f'>SO, ~ vect-ov-
rd
2. ~me~ ten s()y ~ tenSD'<"
eqn 1.1
~OT" T"\,--,ond_ .
: ~ P-i e't p~: C.Orc"lpone'f'-1,-S ""I R.T.
-1,q ...,,... •he 't,asics set,, { ~ 1
no-te, { e-c.~ -
'--
C>'r-t-'no 'nor ma I b a g; s set-
4 ort"'ro - perpendicular ISi nc::.rma\ = 1...
~you \<.nO"'-.J "V'-.JY)Qt- a basis ·,s ~
// "i f
any i nqex, ,n O 8) ng\-e -rerm ,s repea--\--ed ·rt- H1)p \ i e.S Svrnrna-\-,ori
over +nai- \ndex. ",
'3
P-= ~ P-ie~ : P➔ ~ -:. p,~,-+ p~e2. '\' R e~ eqn.1,'2...
'-""" _.-:: I -----. ~ ~ '-"' 3 .....,....
·• cl()-t- prodvcl cit: ;-'N c:,
vec\:-ors is a sco \O r
• · d t>-t- prc,d,vo-\- ~
~·~=;;A-, v,.., "-l2.V2.,.. u..3 v?> peperrj icu\.o.-r vectocs
:. (u."'
~i) • ( vj ~.J) ; '5 -z:_ey- C>
• do-t proovo- of- o.
vn;+- v-e.c~o::::,r b 'f ;-tsse\f
;-s "1.
►
, ~i-he pnys ico\ in-+e rl>re tctt'to n o+ do+ p,..od vc\- ~g i,..-o. lec+\o n o+ ot-.e
'VC:c.-t-'or Orrt"O Ono\- l"'e\'"
t •
.
e.'lt .
_r
o"-. I .,..
-f c, an <::>r- t-n<:> Y"\orm a\ oo.S \s ~ ( '?> vt"'~+ 'lec+ of'9 \ \ to eac'r ic,t'n e)
L..._ e · • e~-:: c;f__. · 9ICI\.J\ 1.Gl,
~ \J "')
dot- pc oduc t- ~ \(.,....o nec\c. .ev- de\"+ o
e'lal uo\e 'o~ ccnt r-oc -hor i
i . e ·. s~ \t, nq ;. 'ne t nct\ Ce!:;>
eqvo L to · eoc not- ne, ~
drop p·•n g -tre \c.'f'"o-neG\(e, de\-t Q
. ' - 1s•."-Jo v
· ~o+ a-n OY" c..o"" ve,.Y" "+; o<'~ : f>
· k .r ovte- c¥-ev - de l to:
·d ~ pro dv c.-r
k-OY "\eC. ~ de\+ a b dov pY-00\...>C+
~- 'X, -: ( \!)~ • ~) • ( '-'j ~)
,,
''co~ +-rac T1<; :>n .,,
L, drof > -\:he de.\+ a ~ set
+re 1nd,c ~s -to~- +r.<:" " sam e
4 re.m a,
te< o"-: I 4J ~I'!= 0
-foy- p~~
, .. , • A - ':,pm oH
""' !, " " . "T 40
c1: P s RSno\'.>i,..o (o
13 Nc:,v 2023
• G ca.tors - rna gnih.Jde, s Peed
L-, h.oro-r"i bY""I '. IX , (3 / Y
vect:-oy- - or,-en-to-t-i o.- ~,..,.~ mag.--..·1--\--\..<le L;"'-e
L, u
.......,..__ I ~ ., U (r---r~"
UJ~ ,J
tensor ·. o Y'T"agn...i +vde \Ni\:-n !WO di-rections
~ .6., ~, A C~ca~)
t e.GY"\-0,i,c.o.,~ -e-ve~n·, ~ ·,s o. -+-e~ OY-
ot-"'- ~ ~r:g.e.. -bensor - ? Scalar-
st
I deq~ te.f'>SO, ~ vect-ov-
rd
2. ~me~ ten s()y ~ tenSD'<"
eqn 1.1
~OT" T"\,--,ond_ .
: ~ P-i e't p~: C.Orc"lpone'f'-1,-S ""I R.T.
-1,q ...,,... •he 't,asics set,, { ~ 1
no-te, { e-c.~ -
'--
C>'r-t-'no 'nor ma I b a g; s set-
4 ort"'ro - perpendicular ISi nc::.rma\ = 1...
~you \<.nO"'-.J "V'-.JY)Qt- a basis ·,s ~
// "i f
any i nqex, ,n O 8) ng\-e -rerm ,s repea--\--ed ·rt- H1)p \ i e.S Svrnrna-\-,ori
over +nai- \ndex. ",
'3
P-= ~ P-ie~ : P➔ ~ -:. p,~,-+ p~e2. '\' R e~ eqn.1,'2...
'-""" _.-:: I -----. ~ ~ '-"' 3 .....,....
·• cl()-t- prodvcl cit: ;-'N c:,
vec\:-ors is a sco \O r
• · d t>-t- prc,d,vo-\- ~
~·~=;;A-, v,.., "-l2.V2.,.. u..3 v?> peperrj icu\.o.-r vectocs
:. (u."'
~i) • ( vj ~.J) ; '5 -z:_ey- C>
• do-t proovo- of- o.
vn;+- v-e.c~o::::,r b 'f ;-tsse\f
;-s "1.
►
, ~i-he pnys ico\ in-+e rl>re tctt'to n o+ do+ p,..od vc\- ~g i,..-o. lec+\o n o+ ot-.e
'VC:c.-t-'or Orrt"O Ono\- l"'e\'"
t •
.
e.'lt .
_r
o"-. I .,..
-f c, an <::>r- t-n<:> Y"\orm a\ oo.S \s ~ ( '?> vt"'~+ 'lec+ of'9 \ \ to eac'r ic,t'n e)
L..._ e · • e~-:: c;f__. · 9ICI\.J\ 1.Gl,
~ \J "')
dot- pc oduc t- ~ \(.,....o nec\c. .ev- de\"+ o
e'lal uo\e 'o~ ccnt r-oc -hor i
i . e ·. s~ \t, nq ;. 'ne t nct\ Ce!:;>
eqvo L to · eoc not- ne, ~
drop p·•n g -tre \c.'f'"o-neG\(e, de\-t Q
. ' - 1s•."-Jo v
· ~o+ a-n OY" c..o"" ve,.Y" "+; o<'~ : f>
· k .r ovte- c¥-ev - de l to:
·d ~ pro dv c.-r
k-OY "\eC. ~ de\+ a b dov pY-00\...>C+
~- 'X, -: ( \!)~ • ~) • ( '-'j ~)
,,
''co~ +-rac T1<; :>n .,,
L, drof > -\:he de.\+ a ~ set
+re 1nd,c ~s -to~- +r.<:" " sam e
4 re.m a,
te< o"-: I 4J ~I'!= 0
-foy- p~~
