Divergence and surfaces integral, very hard

[matt222]

Homework Statement

A vector field h is described in cylindrical polar coordinates by ( h equation attached )
where i, j, and k are the unit vectors along the Cartesian axes and
(er) is the unit vector (x/r) i+(y/r) j
Calculate (1) by surface integral h through the closed surface bounded by the cylinders r=a and r=2a and the planes z=-a*pi/2 to +a*pi/2
and (2) by divergence theorem.

Homework Equations

The Attempt at a Solution

1- I draw the equation and I found it is like hollow tube cylinder with radius of r of the inner surface and 2r for the outer surface
2-I found there are 6 surfaces

a- S1 pointing into -z direction
b- S2 pointing into +z direction
c- S3 pointing into -Y direction
d- S4 pointing into -x direction
e- S5 surface of the outer tube r= 2a
f- S6 surface of the inner tube r=a


I solved the 6 surfaces and the outcome

S1=0

S2=sin(lamda*a*pi/2)*pi*a^2/4

S3=y/lamda*sin(a*pi*lmda/2)

S4=x/lamda*sin(a*pi*lmda/2)

S5= pi/2*lamda[2sin(a*pi*lmda/2)-2acos(a*pi*lmda/2)+2a]

S6=pi/2*lamda[sin(a*pi*lmda/2)-acos(a*pi*lmda/2)+a]

adding them all up and compared them with divergence which I got are not the same, could anyone find out what mistake I did

the divergence I got was

3a*pi/4lamda*sin(a*pi*lmda/2)+7a^2*pi/6*cos(a*pi*lmda/2)+3a*pi/4sin(a*pi*lmda/2)+
7a^2*pi/6

which is not the same with what I got

 

[Coto]

It's very hard to see what you may have done wrong when you've only shown your final result, however just taking a look at what you took for your surfaces:

You mentioned you had 6 surfaces? If you're dealing in cylindrical coordinates I see only 4,

S1: The top of the hollow tube (pointing in the positive \hat{e_z} direction.)
S2: The bottom of the hollow tube (pointing in the negative \hat{e_z} direction.)
S3: The outside of the cylinder r=2a (pointing radially outwards in the \hat{e_r} direction.)
S4: The inside of the cylinder r=a (pointing radially inwards in the \hat{e_r} direction.)

 

[matt222]

what about the other two surfaces which is pointing in -y and -x direction

 

[Coto]

All the surfaces are accounted for. You have two planes which have normal vectors oriented with the z-axis.

You have two cylinders, one twice the radius of the other. The surfaces of cylinders have normal vectors which point radially.

What other surfaces do you have?

 

[matt222]

really lost don't know where is my mistake