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
Problem:
Compute the Gradient and Laplacian of the function
Solution:
cossincos
T ∂T
∂r 1
∂T
∂θ
1
sin
∂T
∂φ
T cos sin cos sin cos cos 1
sin sin sin
cos sin cos sin cos cos sin
T ∙
T 1 ∂ ∂T
∂r ∂r 1
sin
T 1 ∂
1
∂r cos sin cos
sin
1
sin
∂
sin sin
∂φ
1 ∂ ∂r cos sin cos 1
sin
1
sin
∂
sin
∂φ
∂ ∂T
sin
∂θ ∂θ 1
sin
∂ T
∂φ
∂
sin sin cos cos
∂θ
∂
sin sin cos cos
∂θ
1 1
2 cos sin cos
sin 2 sin cos cos cossin cos
1
sin cos
1
sin 2sincos2sin cos2 sincoscos cossin cos
cos
1
sin sin cos cos cos 0
Problem:
Check the divergence theorem for the function , using spherical volume of radius
, centered at the origin.
Divergence theorem state:
∙
V ∙
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