Chapter One: Vector Analysis The use of vectors and vector analysis ...

Electromagnetic Theorem
(Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department
Evidently, gradient have the special property
that their line integral are path independent.
Corollary(1):
∙ is independent of path taken from to .
Corollary(2):
∮ ∙
0 since the beginning and end points are identical, and hence 0.
Example:
Let
, and take point to be the origin 0 , 0 , 0 and the
point 2 , 1 , 0. Check
the fundamental theorem for gradients.
Althoughh the integral is independent of path, we must pick specific path in order to
evaluate it.
Let’s go out along the
(step ) and then up (step );
The fundamental theorem of Gradient state:
0 ;
T y 2
∙
2
⟹
2
(i)
∙ 2
0 ;
0
⟹
T
∙ 0
T
∙ 0