Chapter 10: Q8MP (page 535)

Evaluate $\nabla . e_{u}, \nabla \times e_{v}, \nabla^{2} l n u$

Short Answer

Expert verified

The required values are given below.

$\nabla \cdot {\hat{e}}_{u} = \frac{1}{u} \nabla \times {\hat{e}}_{v} = \frac{\sqrt{2 v - v^{2}}}{\frac{\partial}{\partial u}} (\frac{u}{\sqrt{2 v - v^{2}}} . 1) {\hat{e}}_{z} = \frac{1}{u} {\hat{e}}_{z} \nabla^{2} \ln u = 0$

Step by step solution

Given Information

The given values are mentioned below.

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

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 scale factors are given below.

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

Find the value of $\nabla . {\hat{e}}_{u}$

$\nabla . {\hat{e}}_{u} = \frac{1}{u} \frac{\partial}{\partial u} (u . 1) \nabla . {\hat{e}}_{u} = \frac{1}{u}$

Find the value of curl ${\hat{e}}_{v}$ .

$\nabla \times V = \frac{1}{h_{u} h_{v}} (\frac{\partial}{\partial u} (h_{v} V_{v}) - \frac{\partial}{\partial v} (h_{u} V_{u})) {\hat{e}}_{z} \nabla \times {\hat{e}}_{v} = \frac{\sqrt{2 v - v^{2}}}{\frac{\partial}{\partial u}} (\frac{u}{\sqrt{2 v - v^{2}}} . 1) {\hat{e}}_{z} = \frac{1}{u} {\hat{e}}_{z}$

Find the value of Laplacian.

$\begin{array}{rcl} \nabla^{2} U & = & \frac{1}{h_{u} h_{v}} (\frac{\partial}{\partial u} (\frac{h_{v}}{h_{u}} \frac{\partial U}{\partial u}) + \frac{\partial}{\partial v} (\frac{h_{u}}{h_{v}} \frac{\partial U}{\partial v})) \\ \nabla^{2} \ln u & = & \frac{1}{u} \frac{\partial}{\partial u} (u \frac{\partial}{\partial u} \ln u) \\ = & 0 \end{array}$

The required values are given below.

$\begin{array}{rcl} \nabla . {\hat{e}}_{u} & = & \frac{1}{u} \\ \nabla \times {\hat{e}}_{u} & = & \frac{\sqrt{2 v - v^{2}}}{\frac{\partial}{\partial u}} (\frac{u}{\sqrt{2 v - v^{2}}}) {\hat{e}}_{z} = \frac{1}{u} {\hat{e}}_{z} \\ \nabla^{2} U & = & \frac{1}{h_{u} h_{v}} (\frac{\partial}{\partial u} (\frac{h_{v}}{h_{u}} \frac{\partial U}{\partial u})) + \frac{\partial}{\partial v} (\frac{h_{u}}{h_{v}} \frac{\partial U}{\partial v}) \\ \nabla^{2} \ln u & = & 0 \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Evaluate $\nabla . e_{u}, \nabla \times e_{v}, \nabla^{2} l n u$

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

Oscillations

Medical Physics

Atoms and Radioactivity

Geometrical and Physical Optics

Particle Model of Matter

Astrophysics

Study anywhere. Anytime. Across all devices.

Evaluate ∇.eu,∇×ev,∇2lnu

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

Oscillations

Medical Physics

Atoms and Radioactivity

Geometrical and Physical Optics

Particle Model of Matter

Astrophysics

Study anywhere. Anytime. Across all devices.

Evaluate $\nabla . e_{u}, \nabla \times e_{v}, \nabla^{2} l n u$