탄성체2,황찬규 Fall, 2007 Changyu Hwang Vector and Tensor Analysis
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탄성체2,황찬규
Fall, 2007
Changyu Hwang
Vector and Tensor Analysis
탄성체2,황찬규
Vector Calculus 연습
탄성체2,황찬규
탄성체2,황찬규
(from Maze’s book)
탄성체2,황찬규
(from Maze’s book)
탄성체2,황찬규
탄성체2,황찬규
(from Maze’s book)
탄성체2,황찬규
(from Maze’s book)
탄성체2,황찬규
두개의 막대기를 직각으로 놓고는 반시계 방향으로돌리면, 평면을 뚫고 올라가는 힘.
임의의 평면의 Normal 방향 찾을 때 유용
Reminder !
탄성체2,황찬규
SIMPLY
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Permutation symbol을 이용한 cross product 표시
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연습: 다음 식을 증명하시오.
Permutation symbol sign rule
탄성체2,황찬규
SolutionSolution
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?
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m kjqi q
(first)(second)-(outer)(inner)(첫번째*두번째)- (밖에서*안으로)
증명은어떻게 하~나?
탄성체2,황찬규
(from Maze’s book)증명
탄성체2,황찬규
(from Maze’s book)연습문제
탄성체2,황찬규
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다음 식을 Indicial notation으로 풀어쓰시오.
Ak
탄성체2,황찬규
Ak
SOLUTIONS
탄성체2,황찬규
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탄성체2,황찬규
탄성체2,황찬규
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각자 연습
탄성체2,황찬규
각자 연습
탄성체2,황찬규
Contravariant vectors
HOW ?
탄성체2,황찬규
Example:
탄성체2,황찬규
Covariant vectors
탄성체2,황찬규
Example:
탄성체2,황찬규
탄성체2,황찬규
The gradient g = ∇y is an example of a covariant tensor.The differential position d = dx is an example of a contravariant tensor.
Contravariant/covariant tensors
1 21 2
nn
x
y y ydy dx dx .... dxx x xˆ ˆˆy d g d
∂ ∂ ∂= + + +∂ ∂ ∂
=∇ =
1 2
1 2
x n
n
y y yg y , ,.....,x x x
d dx ,dx ,.....,dx
∂ ∂ ∂⎢ ⎥= ∇ = ⎢ ⎥∂ ∂ ∂⎣ ⎦⎢ ⎥= ⎣ ⎦
여기서
1 21 2
nn
X
y y ydy dX dX .... dXX X X
ˆˆ ˆy D G D
∂ ∂ ∂= + + +∂ ∂ ∂
=∇ =
.ˆˆ ˆg G, d D↔ ↔ 사이관계를구하는것이목표이다
Scalar
Scalar
탄성체2,황찬규
1st rank Contravariant tensors (vector)
새 좌표계의 미소길이를 구 좌표계의 미소길이로 표시하면,
1 21 2
i i ii n
n
ij j
j
X X XdX dx dx .... dxx x x
X NEWdx dxOLDx
∂ ∂ ∂= + + +∂ ∂ ∂
∂= =∂
iNEW
jOLD
Xx
∂∂
1X
2X3X
NEW
1dX
2dX3dX
1x
2x
3x OLD
1dx
2dx3dx
변환대상이VECTOR
탄성체2,황찬규
1 21 2
i i ii n
nX X XdX dx dx .... dxx x x
∂ ∂ ∂= + + +∂ ∂ ∂
1 1 11 1 2
1 2
2 2 22 1 2
1 2
1 21 2
nn
nn
n n nn n
n
X X XdX dx dx .... dxx x xX X XdX dx dx .... dxx x x
..........
X X XdX dx dx .... dxx x x
∂ ∂ ∂= + + +∂ ∂ ∂∂ ∂ ∂
= + + +∂ ∂ ∂
∂ ∂ ∂= + + +∂ ∂ ∂
In a matrix form1 1 1
1 21 12 2 2
2 21 2
1 2
n
n
n nn n n
n
X X X.....x x xdX dxX X XdX dx.....x x x
..... .......... ..... ..... .....dX dxX X X.....
x x xˆD J d
⎡ ⎤∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎧ ⎫ ⎧ ⎫⎢ ⎥
⎪ ⎪ ⎪ ⎪⎢ ⎥∂ ∂ ∂⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬∂ ∂ ∂⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎩ ⎭∂ ∂ ∂⎢ ⎥
⎢ ⎥∂ ∂ ∂⎣ ⎦
=
Jacobian matrix
Contravariant Tensors (vector)
ii NEW j OLDNEW
jOLD
XdX dxx
− −∂=∂
탄성체2,황찬규
구 좌표계의 길이를 새 좌표계의 길이로 표시하면,
1 21 2
i i ii n
nx x xdx dX dX .... dXX X X∂ ∂ ∂
= + + +∂ ∂ ∂
풀어서 써보면,1 1 1
1 1 21 2
2 2 22 1 2
1 2
1 21 2
nn
nn
n n nn n
n
x x xdx dX dX .... dXX X Xx x xdx dX dX .... dXX X X
.......
x x xdx dX dX .... dXX X X
∂ ∂ ∂= + + +∂ ∂ ∂∂ ∂ ∂
= + + +∂ ∂ ∂
∂ ∂ ∂= + + +∂ ∂ ∂
1st rank Covariant tensors (vector)
탄성체2,황찬규
1 21 2
i i ii n
nx x xdx dX dX .... dXX X X∂ ∂ ∂
= + + +∂ ∂ ∂
를 원식에 대입하면,
1 21 2
nn
y y ydy dx dx .... dxx x x∂ ∂ ∂
= + + +∂ ∂ ∂
1 1 11 2
1 1 2
2 2 21 2
2 1 2
1 21 2
nn
nn
n n nn
n n
y x x xdy dX dX dXx X X X
y x x x dX dX dXx X X X
......
y x x x dX dX dXx X X X
⎛ ⎞∂ ∂ ∂ ∂= + + ⋅⋅+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂ ∂+ + + ⋅⋅+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂ ∂+ + + ⋅⋅+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
1 1 11 1 2
1 2
2 2 22 1 2
1 2
1 21 2
nn
nn
n n nn n
n
x x xdx dX dX .... dXX X Xx x xdx dX dX .... dXX X X
.......
x x xdx dX dX .... dXX X X
∂ ∂ ∂= + + +∂ ∂ ∂∂ ∂ ∂
= + + +∂ ∂ ∂
∂ ∂ ∂= + + +∂ ∂ ∂
1 2 ndX ,dX ,...,dX 에대하여정리하면
탄성체2,황찬규
1 21
1 1 1 2 1
1 22
2 1 2 2 2
1 2
1 2
1 2
1 21
n
n
n
n
nn
n n n n
n
k k k nk
x y x y x ydy dXX x X x X x
x y x y x y dXX x X x X x
......
x y x y x y dXX x X x X x
x y x y x yX x X x X x=
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂= + +⋅⋅+⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
+ + +⋅⋅+⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ + +⋅⋅+⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
= + +⋅⋅+⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
nk
X
dX
ˆˆ ˆy D G D=∇ ⋅ = ⋅
∑
1 2
1 2
1 2
1 2
n
i i i i n
n
ni i i
x y x y x yGX x X x X x
x x xˆ ˆ ˆg g gX X X
∂ ∂ ∂ ∂ ∂ ∂= + +⋅⋅+∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂= + +⋅⋅+∂ ∂ ∂
Cartesian Coordinate에서는 둘이 같다.
iOLD
i NEW j OLDjNEW
OLDX x
NEW
xG gX
xy yX
− −∂
=∂
∂∇ = ∇
∂
탄성체2,황찬규
탄성체2,황찬규
2nd rank Covariant tensors
예: 4차원=시간 + 3차원공간
탄성체2,황찬규
탄성체2,황찬규
Notice that each component of the new metric array is a linear combination of the old metric components, and the coefficients are the partials of the old coordinates. with respect to the new. Arrays that transform in this way are called covariant tensors
( ) ( )y x
x xg gy y
μ ν
αβ μνα β∂ ∂
=∂ ∂
탄성체2,황찬규
if we define an array Aμν with the components (dxμ/ds)(dxν/ds) where s denotes a path parameter along some particular curve in space, then equation (2) tells us that this array transforms according to the rule
This is very similar to the previous formula, except that the partial derivatives are of the new coordinates with respect to the old. Arrays whose components transform according to this rule are called contra-variant tensors.
( ) ( )y x
y yA Ax x
α βαβ μν
μ ν∂ ∂
=∂ ∂
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