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

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Electromagnetic Theorem (Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department The curl is a vector, so we can take its divergence and curl: 4‐ Divergence of curl: ∙ 5‐ Curl of Curl: Let’s consider them one at a time: 1) ∙ ∙ ∙ This object, which we write is called the Laplacian of . Note that the Laplacian of a scalar is a scalar. Occasionally, we shall speak of the Laplacian of a vector, . 2) The curl of a gradient is always zero: 0 This is an important fact, which we shall use repeatedly. 3) ∙ for some reason seldom occurs in physical applications. Notice that ∙ is not the same as the Laplace of vector: ∙ ∙ 4) The Divergence of curl ∙ , like the curl of gradient, is always zero: ∙ 0 5) From definition of ∙ We could go through a similar ritual to work out third derivative, but fortunately second derivative suffice for practically all physical applications. 18
Electromagnetic Theorem (Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department Problem: Calculate the Laplacian of the following functions: a) 234 b) sin sin sin c) sin 4 cos 3 d) 3 2 a)) b)) 2 ; 0 ⟹ 2 sin sin sin ⟹ 3 sin sin sin 3 c)) d)) 25 ; 16 ; 9 ⟹ 0 2 ; 0 ⟹ 2 0 ; 6 ⟹ 6 0 ⟹ 0 2 6 Problem: Prove that the divergence of a curl is always zero. ∙ ∙ ∙ By equality of cross‐derivative: ∙ 0 19
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Chapter One: Vector Analysis The use of vectors and vector analysis ...

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