Q8MP Evaluate 鈭�.eu,鈭嚸梕v,鈭�2lnu... [FREE SOLUTION]

Chapter 10: Q8MP (page 535)

Evaluate $鈭� . e_{u}, 鈭� 脳 e_{v}, {鈭�}^{2} l n u$

Short Answer

Expert verified

The required values are given below.

$鈭� 路 {\hat{e}}_{u} = \frac{1}{u} 鈭� 脳 {\hat{e}}_{v} = \frac{\sqrt{2 v - v^{2}}}{\frac{鈭�}{鈭� u}} (\frac{u}{\sqrt{2 v - v^{2}}} . 1) {\hat{e}}_{z} = \frac{1}{u} {\hat{e}}_{z} {鈭�}^{2} \ln u = 0$

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 $鈭� . {\hat{e}}_{u}$

$鈭� . {\hat{e}}_{u} = \frac{1}{u} \frac{鈭�}{鈭� u} (u . 1) 鈭� . {\hat{e}}_{u} = \frac{1}{u}$

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

$鈭� 脳 V = \frac{1}{h_{u} h_{v}} (\frac{鈭�}{鈭� u} (h_{v} V_{v}) - \frac{鈭�}{鈭� v} (h_{u} V_{u})) {\hat{e}}_{z} 鈭� 脳 {\hat{e}}_{v} = \frac{\sqrt{2 v - v^{2}}}{\frac{鈭�}{鈭� u}} (\frac{u}{\sqrt{2 v - v^{2}}} . 1) {\hat{e}}_{z} = \frac{1}{u} {\hat{e}}_{z}$

Find the value of Laplacian.

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

The required values are given below.

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

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Evaluate $鈭� . e_{u}, 鈭� 脳 e_{v}, {鈭�}^{2} l n u$

Short Answer

Step by step solution

Given Information

Definition of a covariant vector

Find the values

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91影视

Evaluate 鈭�.eu,鈭�脳ev,鈭�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

Wave Optics

Atoms and Radioactivity

Classical Mechanics

Astrophysics

Magnetism

Quantum Physics

Study anywhere. Anytime. Across all devices.

Evaluate $鈭� . e_{u}, 鈭� 脳 e_{v}, {鈭�}^{2} l n u$