Example find the curl for the function 2 ˆ ρ k ρ f

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Example : Find the curl for the function 2 ˆ - = ρ k ρ F r , where k is a constant and ρ is the magnitude of the position vector in xy -plane. 1.5.2. Second order spatial derivatives 1.5.2.1. The Laplacian The Laplacian of a scalar f is + + = 3 3 2 1 3 2 2 1 3 2 1 1 3 2 1 3 2 1 2 1 q f h h h q q f h h h q q f h h h q h h h f f (1.42) In cylindrical coordinates it becomes 2 2 2 2 2 2 1 1 1 1 1 1 1 z f f f z f z f f f + + = + + = φ ρ ρ ρ ρ ρ ρ φ ρ φ ρ ρ ρ ρ (1.43) In these two equations, replacing scalar f by vector F r results in the Laplacian of vector F r . Exercise: Express equation (1.42) in (i) rectangular and (ii) spherical coordinates. 1.5.2.2. The divergence of the curl of a vector From equations (1.38) and (1.40), one gets
12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) - + - + - = × 2 1 1 1 2 2 3 3 2 1 1 3 3 3 1 1 2 3 2 1 3 2 2 2 3 3 1 3 2 1 1 1 1 q h F q h F q h h h q h F q h F q h h h q h F q h F q h h h F r (1.44) Exercise: Express equation (1.44) in (i) Cartesian, (ii) cylindrical and (ii) spherical coordinates. 1.6. Integral calculus 1.6.1. Line, surface and volume integrals The most important integrals in electromagnetism are the line (path) integral, surface integral (or flux) and volume integral. 1.6.1.1. Line integral The line integral of a vector function along a prescribed path between points a and b is b a l r r d F where we take the dot product of the vector F r and the elemental length l r d at each point on the path, with l r d being tangential to the path. If the path is closed then the integral is written as cl d l r r F In general these integrals depend on the path taken. There is a special class of vectors for which the line integral does not depend on the path connecting the two points a and b . A force having this property is known as a conservative quantity, and its line integral, which is the work done by it, is independent of he path taken. Example 1 : Find the line integral of a vector 2 ˆ - = cr r v r , with c a constant, from r = r 1 to r = r 2 . Soln .: In spherical coordinates the displacement is φ θ d ˆ d ˆ d ˆ d φ θ r + + = r r l r . Then - = = + + = - - - - 2 1 2 2 2 2 1 1 d d sin ˆ ˆ d ˆ ˆ d ˆ ˆ d 2 1 2 1 2 1 2 1 2 r r c r cr r cr r cr r cr r r r r φ φ θ θ φ θ θ φ r θ r r r v l l 1 l r r where the rules for dot products of unit vectors have been applied.

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