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.

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

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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

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