Chapter One: Vector Analysis The use of vectors and vector analysis ...
Chapter One: Vector Analysis The use of vectors and vector analysis ...
Chapter One: Vector Analysis The use of vectors and vector analysis ...
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Electromagnetic <strong>The</strong>orem<br />
(Dr. Omed Ghareb Abdullah) University <strong>of</strong> Sulaimani –College <strong>of</strong> Science – Physics Department<br />
Example(2):<br />
(a). find the length <strong>of</strong> each <strong>of</strong> the curves: 3,<br />
<br />
,<br />
4 2<br />
z cons tan t.<br />
<br />
(b). Calculate the area <strong>of</strong> the surfaces area defined by: z 1, 1 3 , 0 <br />
4<br />
(c). determine the volume <strong>of</strong> the region defined by: 1 r 3 ,<br />
2<br />
<br />
, <br />
2 3 6 2<br />
Solution:<br />
(a).<br />
dl d<br />
<br />
<br />
dl <br />
/ 2<br />
<br />
d<br />
3 ( / 2 <br />
/ 4)<br />
<br />
4<br />
/ 4<br />
3<br />
l unit<br />
length<br />
(b)<br />
ds d<br />
d<br />
<br />
3 / 4<br />
1 4<br />
3 /<br />
2<br />
<br />
ds <br />
d<br />
d<br />
( ) ( )<br />
4 <br />
2<br />
1 0<br />
4<br />
1<br />
0<br />
<br />
S <br />
unit<br />
<strong>of</strong><br />
area<br />
(c).<br />
dv r<br />
2<br />
sin<br />
dr d<br />
d<br />
<br />
<br />
dv <br />
3 2<br />
/ 3 / 2<br />
<br />
1 / 2<br />
/ 6<br />
3<br />
r<br />
sin<br />
dr d<br />
d<br />
<br />
3<br />
1<br />
v (27 1)<br />
(cos<br />
/ 2 cos 2<br />
/ 3) ( / 2 / 6)<br />
3<br />
1<br />
<br />
13<br />
v (26) (0<br />
0.5) (<br />
) v unit <strong>of</strong><br />
3<br />
3<br />
9<br />
r<br />
2<br />
volume<br />
3<br />
1<br />
( cos<br />
)<br />
2<br />
/ 3<br />
/ 2<br />
( )<br />
/ 2<br />
/ 6<br />
Example(3):<br />
Find the gradient <strong>of</strong> the following scalar functions:<br />
a. U 4 x z<br />
2 3 y z , b. 2 ( z 2 2<br />
W 1)<br />
cos<br />
, c. V r cos cos<br />
.<br />
(a). <br />
U (<br />
<br />
x<br />
aˆ<br />
x<br />
<br />
<br />
y<br />
aˆ<br />
y<br />
<br />
<br />
z<br />
aˆ<br />
)(4xz<br />
z<br />
2<br />
3yz)<br />
<br />
4 z<br />
2<br />
aˆ<br />
x<br />
3z aˆ<br />
y<br />
(8xz<br />
3y)<br />
aˆ<br />
z<br />
(b).<br />
<br />
<br />
W ( aˆ<br />
<br />
<br />
1 <br />
aˆ<br />
<br />
<br />
<br />
<br />
z<br />
aˆ<br />
)(2 (<br />
z<br />
1) cos)<br />
2( z<br />
1) cos<br />
aˆ<br />
2 ( z<br />
1) sin<br />
aˆ<br />
4 z cos<br />
ˆ<br />
2<br />
2<br />
2<br />
z <br />
<br />
a z<br />
56
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