Begin2.DVI

I8-13. V =^1
2

kx^2 withF~ =mddt^2 ~r 2. UseF~ = gradV =−kxˆe 1. That is, if the spring

is stretched in the positive direction a distancex, then the restoring force is in the

negative direction and proportional to the displacement. This gives the equation of

motion for the spring mass system as

m

d^2 x

dt^2 +kx= 0 or

d^2 x

dt^2 +ω

(^2) x= 0, ω (^2) =k
m
I8-14.
F~(x+ ∆x,y,z) =F~(x,y,z) +∂F~
∂x
∆x+h.o.t.
F~(x,y+ ∆y,z) =F(x,y,z) +∂F~
∂y
∆y+h.o.t.
F~(x,y,z+ ∆z) =F~(x,y,z) +∂F~
∂z∆z+h.o.t.
whereh.o.t.denotes ”higher order terms” which are neglected.
The flux in thex-direction on face CGBF isF~·(−ˆe 1 ) ∆S=−F 1 ∆y∆zand the flux
in thex-direction of the face DHAE is
(F~+∂
F~
∂x
∆x)·(ˆe 1 ) ∆S= (F 1 +∂F^1
∂x
∆x) ∆y∆z
The flux in they-direction on face DCBA isF~·(−ˆe 2 ) ∆S=−F 2 ∆x∆zand the flux in
they-direction on face HGEF is
(F~+∂
F~
∂y
∆y)·(ˆe 2 ) ∆S= (F 2 +∂F^2
∂y
∆y) ∆x∆z
The flux in thez-direction on face AEFB isF~·(−ˆe 3 ) ∆S=F 3 ∆x∆y and the flux in
thez-direction on face HGCD is
(F~+
∂F~
∂z∆z)·
ˆe 3 ∆S= (F 3 +∂F^3
∂z ∆z) ∆x∆y
Add the flux over each surface and show
Total Flux=
(
∂F 1
∂x
+∂F^2
∂y
+∂F^3
∂z
)
∆x∆y∆z
so that
Flux
V olume
=∂F^1
∂x
+∂F^2
∂y
+∂F^3
∂z
= divF~
I8-15. (i)divF~= 2yz− 2 x, curlF~=~ 0
(iii)divF~= 2y− 2 z curlF~=~ 0
Solutions Chapter 8