Chapter 10: Q7MP (page 535)

Write $\nabla U, \nabla V, \nabla^{2} U$

Short Answer

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$T h e r e q u i r e d v a l u e s a r e g i v e n b e l o w . \nabla U = \frac{\partial U}{\partial u} {\hat{e}}_{u} + \frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial U}{\partial v} {\hat{e}}_{v} \nabla \cdot V = \frac{1}{u} \frac{\partial}{\partial u} (u V_{u}) + \frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial}{\partial v} V_{v} \nabla^{2} U = \frac{1}{u} \frac{\partial}{\partial u} (u \frac{\partial U}{\partial u} + \frac{\partial}{h_{u} h_{v}} \frac{h_{v}}{\partial u} \frac{\partial U}{\partial u}) + \frac{1}{h_{u} h_{v}} \frac{\partial}{\partial v} (\frac{h_{u}}{h_{v}} \frac{\partial U}{\partial v} (\frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial U}{\partial v}))$

Step by step solution

Given Information

The given values are mentioned below.

Definition of a covariant vector

Components of a covariant vector or cotangent vector (commonly abbreviated as co vector) co-vary as the basis changes.

Find the values

The given values are mentioned below.

$x = u (1 - v) y = u \sqrt{2 v - v^{2}}$

The scale factors are given below.

$h_{u} = 1 h_{v} = \frac{u}{\sqrt{2 v - v^{2}}}$

Find the value of $\nabla U$ .

$\nabla U = \frac{1}{h_{u}} \frac{\partial U}{\partial u} {\hat{e}}_{u} + \frac{1}{h_{v}} \frac{\partial U}{\partial v} {\hat{e}}_{v} \nabla U = \frac{\partial U}{\partial u} {\hat{e}}_{u} + \frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial U}{\partial v} {\hat{e}}_{v}$

Find the value of $\nabla . V$

$\nabla . V = \frac{1}{h_{u} h_{v}} \frac{\partial}{\partial u} (h_{v} V_{u}) + \frac{1}{h_{u} h_{v}} \frac{\partial}{\partial V} (h_{u} V_{v}) \nabla . V = \frac{1}{u} \frac{\partial}{\partial u} (u V_{u v}) + \frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial}{\partial v} (V_{v})$

Find the value of $\nabla^{2} U$ .

$\nabla^{2} U = (\frac{1}{u} \frac{\partial}{\partial u} (u \frac{\partial U}{\partial u}) + \frac{\partial}{h_{u} h_{v}} \frac{h_{v}}{\partial u} \frac{\partial U}{\partial u}) + \frac{1}{h_{u} h_{v}} \frac{\partial}{\partial v} (\frac{h_{u}}{h_{v}} \frac{\partial U}{\partial v} (\frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial U}{\partial v}))$

The required values are given below.

$\nabla U = \frac{\partial U}{\partial u} {\hat{e}}_{u} + \frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial U}{\partial v} {\hat{e}}_{v} \nabla \cdot V = \frac{1}{u} \frac{\partial}{\partial u} (u V_{u}) + \frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial}{\partial v} V_{v} \nabla^{2} U = \frac{1}{u} \frac{\partial}{\partial u} (u \frac{\partial U}{\partial u}) + \frac{\partial}{h_{u} h_{v}} \frac{h_{v}}{\partial u} \frac{\partial U}{\partial u} + \frac{1}{h_{u} h_{v}} \frac{\partial}{\partial v} (\frac{h_{u}}{h_{v}} \frac{\partial U}{\partial v} (\frac{\sqrt{2 v - v^{2}}}{u} \frac{\partial U}{\partial v}))$

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Write $\nabla U, \nabla V, \nabla^{2} U$

Short Answer

Step by step solution

Given Information

Definition of a covariant vector

Find the values

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Most popular questions from this chapter

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Write∇U,∇V,∇2U

Short Answer

Step by step solution

Given Information

Definition of a covariant vector

Find the values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Solid State Physics

Linear Momentum

Force

Oscillations

Quantum Physics

Study anywhere. Anytime. Across all devices.

Write $\nabla U, \nabla V, \nabla^{2} U$