1 Continuum Mechanics Notation Words Symbolic Traditional Cartesian Cartesian Tensor Curvilinear Tensor Defining a point in Euclidean space relative to an origin r r = xi + yj + zk r = x i e i the summation i=1 3 ! is implied r = xx 1 , x 2 , x 3 ( ) i + yx 1 , x 2 , x 3 ( ) j + zx 1 , x 2 , x 3 ( ) k Base vectors Not applicable i • i = 1 j • j = 1 k • k = 1 and i • j = 0 j • k = 0 k • i = 0 e i • e j = ! ij The Kronecker delta, ! ij = 1 if i = j and ! ij = 0 if i ! j . g i = !r !x i = r , i g i • g j = ! i j g i • g j = g ij g i • g j = g ij The Kronecker delta, ! i j = 1 if i = j and ! j i = 0 if i ! j . g i are the covariant base vectors. g i are the contravariant base vectors. The curvilinear coordinates, x i always have superscripts. Scalar - 0 th order tensor a , ! a , ! a , ! a , ! Element of displacement ds = dr = dr • dr dr = dxi + dyj + dzk ds = dx 2 + dy 2 + dz 2 dr = dx i e i ds = dx i dx i dr = dx i g i ds = g ij dx i dx j
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Continuum Mechanics Notation
Words Symbolic Traditional Cartesian Cartesian Tensor Curvilinear Tensor
Defining a point in Euclidean space relative to an origin
r
r = xi + yj + zk
r = xie i the summation
i=1
3
! is implied
r = x x1, x2 , x3( )i+ y x1, x2, x3( )j
+ z x1, x2, x3( )k
Base vectors Not applicable
i • i = 1j• j = 1k • k = 1
and
i • j = 0j• k = 0k • i = 0
e i • e j = !ij The Kronecker delta,
!ij = 1 if
i = j and
!ij = 0 if
i ! j .
g i = !r!x i = r, i
g i • g j = !ij
g i • g j = gij
g i • g j = g ij
The Kronecker delta,
!ij = 1 if
i = j and
! ji = 0 if
i ! j .
g i are the covariant base vectors.
g i are the contravariant base vectors. The curvilinear coordinates,