I7-14. Use the vector identityA~×(B~×C~) =B~(A~·C~)−C~(A~·B~)and show
A~×(B~×C~) =B~(A~·C~)−C~(A~·B~)
B~×(C~×A~) =C~(B~·A~)−A~(B~·C~)
C~×(A~×B~) =A~(C~·B~)−B~(C~·A~)Add both sides to obtain desired result.I7-15.
∂z
∂x=x(^2) +x+ 2, ∂z
∂y=y
(^2) − 3 y+ 2
critical points are where ∂z∂x= 0and∂z∂y= 0simultaneously
critical points(− 2 ,2), (− 2 ,1), (1,2), (1,1)
A=
∂^2 z
∂x^2
= 2x+ 1 =A,
∂^2 z
∂x∂y
= 0 =B,
∂^2 z
∂y^2
= 2y−3 =C, ∆ =AC−B^2
At(2,2),∆ =− 3 < 0 , saddle point
At(− 2 ,1), ∆ = 3> 0 , A < 0 , relative minimum
At(1,2),∆ = 3> 0 , A > 0 , relative minimum
At(1,1),∆ =− 3 < 0 , saddle point
I7-16. ~ω=αˆet+βˆen+γˆeband if
ω׈et=κˆen, ω׈eb=−τˆen, ω׈en=τˆeb−κˆet
show thatα=τ,β= 0,γ=κso thatω=τˆet+κˆeb
Solutions Chapter 7