basic memorizable equations for index notation

Question 1

[VW]x(““means times)

Answer 1

VyWz-VzWy= -[W*V]x

Question 2

(V.W)(. means dot product)

Answer 2

VxWx+VyWy+VzWz= (W.V)

Question 3

[T.V]x (T is actually TAO the tensor )

Answer 3

TxxVx+TxyVy+TxzVz

Question 4

[V.T]x

Answer 4

VxTxx+VyTyx+VzTzx

Question 5

[deltaS]x

Answer 5

eS/ex( e means partial differential)

Question 6

[delta*V]x

Answer 6

(eVz/ey)-(eVy/ez)

Question 7

(delta.V)

Answer 7

(eVx/ex)+(eVy/ey)+(eVz/ez)

Question 8

(V.delta s)

Answer 8

(Vx es/ex)+(Vy es/ey)+(Vz es/ez)

Question 9

delta^2 s

Answer 9

e^2s/ex^2+e^2s/ey^2+e^2s/ez^2(second derivative )

Question 10

[delta.VV]x( remember only last V interfere with x )

Answer 10

e(VxVx)/ex+ e(VyVx)/ey+e(VzVx)/ez

Question 11

[V.deltaV]x

Answer 11

Vx eVx/ex+ Vy eVx/ey+ Vz eVx/ez

Question 12

[delta.T]x

Answer 12

eTxx/ex+ eTyx/ey+eTzx/ez

Question 13

(T:delta v)

Answer 13

Txx eVx/ex+ Txy eVx/ey+ Txz eVx/ez
Tyx eVy/ex+ Tyy eVy/ey+ Tyz eVy/ez
Tzx eVz/ex+ Tzy eVz/ey+ Tzz eVz/ez