Convert surface normal vectors and tangent vectors

Question

According to the book "Physical justification": from theory to implementation. Matt Farr, Greg Humphries ( Ref. , Pp. 86-87),
surface tangent vectors are transformed as common vectors using the transformation matrix M , but
surface normal vectors are transformed using .

I wonder why scaling makes it normal wrong, but does not touch the tangent vector? Why are normals so special?

See drawing from book.

, . .

Answer 1

, , bivectors, , 3D , . , . 4 6 .

/ . i, j, k - , j ^ k, i ^ k i ^ j, Hodge Dual . , .

, , , - , , , , .

. , (1/rt2, 1/rt2, 0) (0,0,1), rt2 = sqrt (2). ,

( 1/rt2 )    ( 0 )    (  1/rt2 )
( 1/rt2 ) X  ( 0 ) =  ( -1/rt2 )
(   0   )    ( 1 )    (   0    )

. (x, y, z) → (x, y/2, z), , (1/rt2, 1/(2 rt2), 0) (0,0,1).

( 1/rt2     )    ( 0 )    (  1/(2 rt2) )
( 1/(2 rt2) ) X  ( 0 ) =  ( -1/rt2     )
(   0       )    ( 1 )    (   0        )

, ( 1/sqrt(5), -2/sqrt(5), 0 ).

, , . . .

Answer 2

, ( , ) . , , .

, ,

, , , .

, . , , , , , . , . , :

, , , .

, .