Explanation Calculation: The standard equation for the gradient in Cartesian coordinates is, ∇ = x ∧ ∂ ∂ x + y ∧ ∂ ∂ y + z ∧ ∂ ∂ z (1) Here, x ∧ is the unit vector along x -axis. y ∧ is the unit vector along y -axis. z ∧ is the unit vector along z -axis. The standard relation between Cartesian and spherical coordinate system is, x = R sin θ cos ϕ y = R sin θ sin ϕ z = R cos θ Where, R = x 2 + y 2 + z 2 θ = tan − 1 [ x 2 + y 2 / z ] ϕ = tan − 1 ( y / x ) Here, R is the radial distance. θ is the zenith angle measured from positive z − axis. ϕ is the azimuth angle measured from positive x − axis. From differential calculus, ∂ ∂ x = ∂ ∂ R ⋅ ∂ R ∂ x + ∂ ∂ θ ⋅ ∂ θ ∂ x + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ x ∂ ∂ y = ∂ ∂ R ⋅ ∂ R ∂ y + ∂ ∂ θ ⋅ ∂ θ ∂ x + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ y ∂ ∂ z = ∂ ∂ R ⋅ ∂ R ∂ z + ∂ ∂ θ ⋅ ∂ θ ∂ z + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ z Substitute ∂ ∂ R ⋅ ∂ R ∂ x + ∂ ∂ θ ⋅ ∂ θ ∂ x + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ x for ∂ ∂ x , ∂ ∂ R ⋅ ∂ R ∂ y + ∂ ∂ θ ⋅ ∂ θ ∂ y + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ y for ∂ ∂ y , ∂ ∂ R ⋅ ∂ R ∂ z + ∂ ∂ θ ⋅ ∂ θ ∂ z + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ z for ∂ ∂ z in equation (1). ∇ = x ∧ ∂ ∂ x + y ∧ ∂ ∂ y + z ∧ ∂ ∂ z ∇ = { x ∧ ( ∂ ∂ R ⋅ ∂ R ∂ x + ∂ ∂ θ ⋅ ∂ θ ∂ x + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ x ) + y ∧ ( ∂ ∂ R ⋅ ∂ R ∂ y + ∂ ∂ θ ⋅ ∂ θ ∂ y + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ y ) + z ∧ ( ∂ ∂ R ⋅ ∂ R ∂ z + ∂ ∂ θ ⋅ ∂ θ ∂ z + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ z ) } (2) Substitute R ∧ sin θ cos ϕ + θ ∧ cos θ cos ϕ − ϕ ^ sin ϕ for x ∧ , R ∧ sin θ sin ϕ + θ ∧ cos θ sin ϕ + ϕ ^ cos ϕ for y ∧ , R ∧ cos θ − θ ∧ sin θ for z ∧ in equation (2). ∇ = { ( R ∧ sin θ cos ϕ + θ ∧ cos θ cos ϕ − ϕ ^ sin ϕ ) ( ∂ ∂ R ⋅ ∂ R ∂ x + ∂ ∂ θ ⋅ ∂ θ ∂ x + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ x ) + ( R ∧ sin θ sin ϕ + θ ∧ cos θ sin ϕ + ϕ ^ cos ϕ ) ( ∂ ∂ R ⋅ ∂ R ∂ y + ∂ ∂ θ ⋅ ∂ θ ∂ y + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ y ) + ( R ∧ cos θ − θ ∧ sin θ ) ( ∂ ∂ R ⋅ ∂ R ∂ z + ∂ ∂ θ ⋅ ∂ θ ∂ z + ∂ ∂ ϕ ⋅ ∂ ϕ ∂ z ) } (3) Differentiate R with respect to x . ∂ R ∂ x = x x 2 + y 2 + z 2 = sin θ cos ϕ Differentiate R with respect to y

7th Edition

ISBN: 8220100663659

Author: ULABY

Publisher: PEARSON

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Question

Chapter 3, Problem 39P

To determine

The Eq. (3.83) for ∇ in spherical coordinates.

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Chapter 3, Problem 38P

Chapter 3, Problem 40P

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