Explanation Given: The gradient of a scalar function is: ∇ → ⋅ f = ∂ f ∂ y y ^ + ∂ f ∂ z z ^ …… (1) Formula used: Write the expression for the rotation of a system. a ¯ = R ′ a …… (2) Here, a ¯ is the matrix after transformation, R ′ is the two dimensional rotation matrix and a is the matrix before transformation. Write the matrix form of rotation matrix of a system. ( y ¯ z ¯ ) = ( cos ϕ sin ϕ − sin ϕ cos ϕ ) ( y z ) …… (3) Calculation: Multiply the matrices in equation (3) and simplify. The first equation of rotation is: y ¯ = y cos ϕ + z sin ϕ …… (4) The second equation of rotation is: z ¯ = − y sin ϕ + z cos ϕ …… (5) Multiply equation (4) with sin ϕ and equation (5) with cos ϕ . y ¯ sin ϕ = y cos ϕ sin ϕ + z sin 2 ϕ , z ¯ cos ϕ = − y cos ϕ sin ϕ + z cos 2 ϕ Add the above equations and simplify. z = y ¯ sin ϕ + z ¯ cos ϕ …… (6) Differentiate equation (6) with respect to y ¯ . ∂ z ∂ y ¯ = sin ϕ …… (7) Differentiate equation (6) with respect to z ¯ . ∂ z ∂ z ¯ = cos ϕ …… (8) Multiply equation (4) with cos ϕ and equation (5) with sin ϕ . y ¯ cos ϕ = y cos 2 ϕ + z cos ϕ sin ϕ , z ¯ sin ϕ = − y sin 2 ϕ + z cos ϕ sin ϕ Subtract the above equations and simplify. y = y ¯ cos ϕ − z ¯ sin ϕ …… (9) Differentiate equation (9) with respect to y ¯ . ∂ y ∂ y ¯ = cos ϕ …… (10) Differentiate equation (9) with respect to z ¯ . ∂ y ∂ z ¯ = − sin ϕ …… (11) Write the expression for divergence of a scalar function with respect to y . ( ∇ → ⋅ f ) ¯ y = ∂ f ∂ y ⋅ ∂ y ∂ y ¯ + ∂ f ∂ z ⋅ ∂ z ∂ y ¯ Substitute ( ∇ → ⋅ f ) y for ∂ f ∂ y , ( ∇ → ⋅ f ) z for ∂ f ∂ z , cos ϕ for ∂ y ∂ y ¯ and sin ϕ for ∂ z ∂ y ¯ in above equation

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Author: David J. Griffiths

Publisher: Cambridge University Press

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Chapter 1.2, Problem 1.14P

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Introduction to Electrodynamics

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