Q12P Do Problem 5 for the coordinate ... [FREE SOLUTION]

Chapter 10: Q12P (page 528)

Do Problem 5 for the coordinate systems indicated in Problems 10 to 13. Parabolic.

Short Answer

Expert verified

The required values are mentioned below.

$鈭� U = \frac{1}{\sqrt{u^{2} + v^{2}}} \frac{鈭� U}{鈭� u} {\hat{e}}_{u} + \frac{1}{\sqrt{u^{2} + v^{2}}} \frac{鈭� U}{鈭� v} {\hat{e}}_{v} + \frac{1}{u v} \frac{鈭� U}{鈭� 蠒} {\hat{e}}_{蠒} 鈭� . V = \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} u v V_{1}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} \sqrt{u^{2} + v^{2} u v V_{2}) + \frac{1}{u v} \frac{鈭� V_{3}}{鈭� 蠒}} {鈭�}^{2} U = \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� v} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{u^{2} v^{2}} \frac{{鈭�}^{2} U}{鈭� 蠒} 鈭� 脳 V = \frac{1}{u v \sqrt{u^{2} + v^{2}}} \{\frac{鈭�}{鈭� v} (u v V_{3}) - \frac{鈭�}{鈭� 蠒} (\sqrt{u^{2} + v^{2} V_{2}})\} {\hat{e}}_{u} + \frac{1}{u v \sqrt{u^{2} + v^{2}}} \{\frac{鈭�}{鈭� 蠒} \sqrt{u^{2} + v^{2}} V_{1}) - \frac{鈭�}{鈭� u} (u v V_{3})\} {\hat{e}}_{v} + \frac{1}{u^{2} + v^{2}} \{\frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} V_{2}) - \frac{鈭�}{鈭� v} (\sqrt{u^{2} + v^{2}} V_{1})\} {\hat{e}}_{蠒}$

Step by step solution

Given Information

The Parabolic.

Definition of cylindrical coordinates.

The coordinate system primarily utilized in three-dimensional systems is the cylindrical coordinates of the system. The cylindrical coordinate system is used to find the surface area in three-dimensional space.

Determining the value of ∇U, ∇.V , ∇2U and ∇×Vand

The scale factors are given below.

$h_{u} = \sqrt{u^{2} + v^{2}} h_{v} = \sqrt{u^{2} + v^{2}} h_{蠒} = u v$

Find $鈭� U$ in Parabolic coordinates.

$鈭� U = \frac{1}{h_{u}} \frac{鈭� U}{鈭� u} {\hat{e}}_{u} + \frac{1}{h_{u}} \frac{鈭� U}{鈭� v} {\hat{e}}_{v} + \frac{1}{h_{u}} \frac{鈭� U}{鈭� 蠒} {\hat{e}}_{蠒} 鈭� U = \frac{1}{\sqrt{u^{2} + v^{2}}} \frac{鈭� U}{鈭� u} {\hat{e}}_{u} + \frac{1}{\sqrt{u^{2} + v^{2}}} \frac{鈭� U}{鈭� u} {\hat{e}}_{v} + \frac{1}{u v} \frac{鈭� U}{鈭� 蠒} {\hat{e}}_{蠒}$

Find the divergence of V.

$鈭� . V = \frac{1}{(u^{2} + v^{2}) u v} \{\frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} u v V_{1}) + \frac{鈭�}{鈭� v} (\sqrt{u^{2} + v^{2}} u v V_{2}) + \frac{鈭�}{鈭� 蠒} ((u^{2} + v^{2}) V_{3})\} 鈭� . V = \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} u v V_{1}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} \sqrt{u^{2} + v^{2} u v V_{2}) + \frac{1}{u v} \frac{鈭� V_{3}}{鈭� 蠒}}$

Find the Laplacian of U.

${鈭�}^{2} U = \frac{1}{(u^{2} + v^{2}) u v} \{\frac{鈭�}{鈭� u} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� v} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{u^{2} v^{2}} \frac{{鈭�}^{2} U}{鈭� 蠒}\} {鈭�}^{2} U = \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� v} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{u^{2} v^{2}} \frac{{鈭�}^{2} U}{鈭� 蠒}$

Find curl of V.

$鈭� 脳 V = \frac{1}{u v \sqrt{u^{2} + v^{2}}} \{\frac{鈭�}{鈭� v} (u v V_{3}) - \frac{鈭�}{鈭� 蠒} (\sqrt{u^{2} + v^{2} V_{2}})\} {\hat{e}}_{u} + \frac{1}{u v \sqrt{u^{2} + v^{2}}} \{\frac{鈭�}{鈭� 蠒} \sqrt{u^{2} + v^{2}} V_{1}) - \frac{鈭�}{鈭� u} (u v V_{3})\} {\hat{e}}_{v} + \frac{1}{u^{2} + v^{2}} \{\frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} V_{2}) - \frac{鈭�}{鈭� v} (\sqrt{u^{2} + v^{2}} V_{1})\} {\hat{e}}_{蠒}$

Hence, the required values are mentioned below.

role="math" localid="1659339435729" $鈭� U = \frac{1}{\sqrt{u^{2} + v^{2}}} \frac{鈭� U}{鈭� u} {\hat{e}}_{u} + \frac{1}{\sqrt{u^{2} + v^{2}}} \frac{鈭� U}{鈭� v} {\hat{e}}_{v} + \frac{1}{u v} \frac{鈭� U}{鈭� 蠒} {\hat{e}}_{蠒} 鈭� . V = \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} u v V_{1}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} \sqrt{u^{2} + v^{2} u v V_{2}) + \frac{1}{u v} \frac{鈭� V_{3}}{鈭� 蠒}} {鈭�}^{2} U = \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� u} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{(u^{2} + v^{2}) u v} \frac{鈭�}{鈭� v} (u v \frac{鈭� U}{鈭� u}) + \frac{1}{u^{2} v^{2}} \frac{{鈭�}^{2} U}{鈭� 蠒} 鈭� 脳 V = \frac{1}{u v \sqrt{u^{2} + v^{2}}} \{\frac{鈭�}{鈭� v} (u v V_{3}) - \frac{鈭�}{鈭� 蠒} (\sqrt{u^{2} + v^{2} V_{2}})\} {\hat{e}}_{u} + \frac{1}{u v \sqrt{u^{2} + v^{2}}} \{\frac{鈭�}{鈭� 蠒} \sqrt{u^{2} + v^{2}} V_{1}) - \frac{鈭�}{鈭� u} (u v V_{3})\} {\hat{e}}_{v} + \frac{1}{u^{2} + v^{2}} \{\frac{鈭�}{鈭� u} (\sqrt{u^{2} + v^{2}} V_{2}) - \frac{鈭�}{鈭� v} (\sqrt{u^{2} + v^{2}} V_{1})\} {\hat{e}}_{蠒}$

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91影视

Do Problem 5 for the coordinate systems indicated in Problems 10 to 13. Parabolic.

Short Answer

Step by step solution

Given Information

Definition of cylindrical coordinates.

Determining the value of ∇U, ∇.V , ∇2U and ∇×Vand

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