- #1

Haths

- 33

- 0

1. Find the surface integral of

**E**. d

**S**where

**E**is a vector field given;

E = yi - xj + 1/3 z

^{3}and S is the surface x

^{2}+ z

^{2}< r

^{2}and 0 < y < b

Well Gauss' theorum would be the place to start to make this easy for me...Already having calculated the divergance of the field to be;

div

**E**= z

^{2}

But my problems came to the paramaterisation of the integral. Assuming that my co-ord. transfer is;

x = r cos(a)

y = y

z = r sin(a)

z

^{2}= r

^{2}sin

^{2}(a)

The jacobian for the cylindrical co-ords I believe being r

^{2}

Hence my integral becomes;

[tex]

$ int int int r^{4} sin^{2}(a) dr da dy$

[/tex]

The limits being b -> 0 / 2PI -> 0 / r -> 0 respectfully

...and I end up with;

1/5 PI b r

^{5}

I have no idea if this is right or not, it 'feels' right, but I'm not confident in the answer, nor if I have made some mistake along the way.

2. Finding the curl of;

[tex]

A(r,t) = a e^{(ip \dot r - i \omega t)}

[/tex]

where a and p are constant vectors...

The only idea I had was to convert it into sines and cosines, but that didn't help me that much.

Cheers,

Haths