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Chapter 10: 15P (page 535)

Do Problem 14 for an orthogonal coordinate system with scale factorsh1,h2,h3,and compare with the Section 9 formulas

Short Answer

Expert verified

The gradient, divergence and curl are obtained.

∇u=1hi∂u∂xie^i∇⋅V=1h1h2h3∂∂xi{h1h2h3Vi}∇2u=1h1h2h3∂∂xi{h1h2h3hi2∂u∂xi}

Step by step solution

01

Given information.

An orthogonal coordinate system is given with scale factorsh1,h2,h3 .

02

Definition of a covariance and contravariance.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant

03

Begin by calculating the gradient.

Calculate the gradient.

∇u=∂u∂xiaiai=gijaj=1hi2ai∇u=∂u∂xiaihi2

Continue the evaluation.

ai=hie^i∇u=1hi∂u∂xie^i

04

Calculate the divergence.

∇⋅V=1g∂∂xi{gVi}g=h1h2h3∇⋅V=1h1h2h3∂∂xi{h1h2h3Vi}

05

Calculate the Laplacian.

∇2u=1h1h2h3∂∂xi{h1h2h3hi2∂u∂xi}

06

Recognize the relation between physical components and contravariance components and substitute.

TheV⋅e^i=hiVidescribes the relation between physical components and contravariance components. Substitute this in earlier equations.

∇⋅V=1h1h2h3{∂∂x1{h2h3V⋅e^1}+∂∂x2{h1h3V⋅e^2}+∂∂x3{h1h2V⋅e^3}}

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