Meccanica razionale - Proofs parte 1

TENSORS

Proof 1 (R3) #u ver

We Stu

Selim

take S F

E

U .

=

: ,

.

St transpose of S

is the

Sji Stji

Sij Stei

Sij Sej ej

ei =

= = · = · Lim(R3) symmetric

WT skew

We

↳ is

BT Lim(3) if W .

ther

Be

ther

B

if is Symmetric = =

= ; ,

, if

If WI

there vector the

exists 1

only

and vector unique

w +

WS is

a T = w

.

.

if there

there

W exist

symmetric

start

We skew unique

by that is

proving w

a

, ?

W

What the representing

about matrix

we say

can element

forex the

This

know the

that that

We transpose

is .

minus

w means .

,

- wji)

(Wij

the diagonal 0

.

are =

or the

take diagonal

that

this

j element

the

If it have in

wil"-Wi means

we :

we

Must 0

be . Wis

- Wie

o

W = Wiz Wizs

0

-

--Wis- Was O &

Now Wie

that Wis

Bet Wa

Wa

-Was =

=

we we = -

,

,

det

I &

WXV en

es

= Vi U3

As regards applied to matrices

using

1

w ,

[c] I

"(1

=

I

Tv] - -

I

0 W3Vzt

]

was Vi WzV3

- - -

WI W

= =

: W3V,

W3 =

- Wils

w

O - , V3

o

Wz W

w Vz

WaVi +

- -

, ,

this

that

We have to unique

still w is

prove WW

To W

w'ER3st

that take

show E

another vector v

w

we =

= +

+

-

, FER3

(w' w) 0

have

But ther +v

we =

- parallel

when the

of vectors

rectors

product

the

We that vector 0

is

have ore

2

seen

for but implies W-W

that

this

take to

parallel

rector

the Is

ex I is

we es

as

. x21GER

w'

=> w =

- (W - w)

with w'

Let's the B2z

w

do 22

e2 0 =

x

:

same =

= -

Since B

<=

baris 0

then

es 22 is =

es a

,

,

W'

So W

= statement

the

have to

We inverse

prove :

I transformation

this linear

ther

WXX is

define 1 a

we = ,

And have to what if

happens

we compute

we

now see - WT -W

1) (wX4)

WX (a transform

w)

(x the

.

u

x = =

+ =

-

+

= =

.

Symmetric

skew

is : TENSORS

PROOF 2 handed

A if transforms

if

linear it right

transformation and

rotation

R only

is a

a handed

into right othonormal

basis basic

athonamal another .

We first rotation

that R is

assume a

RV

RA RTRV V -

M

u =

=

. -

. delta Rei *

Rej

If Li

observe Sij

that Ej

basis

Er

En is we

es = = .

·

a

, , , (A) Sij

If define Mj

from

then

RE Res

and =

RE2

M .

=

we Mr Us = ,

=

,

vectors

the orthomoral

M

=> 42 As are

,

, - 2 x22 23

=

,

-

handed

Now right

have these

to that rectors also

see

we are to

but

that e2)

(Er

if compute

this

compute do to

It have

1 we

we E

means =

·

+ x(z)

(detr)(

Rez)

(M1 (REX

Xuz) Rez zo

uz =

= .

- . ,

x()

(2

rotation

definition of

But by 1

now 23

= =

-

,

I

, &3)

(M basic

right-handed

is

Hr a

=> , , us)

23) (11

and

(2

that othorarwal handed

right

We bas