We just started a section dealing with div/curl/grad in different orthogonal systems... before I get started doing problems involving these operations I wanted to make sure I am dealing with these operation correctly. Our first homework problem is as follows:(adsbygoogle = window.adsbygoogle || []).push({});

In cylindrical coordinates compute:

(1) [tex] \nabla \theta [/tex]

(2) [tex] \nabla r^4 [/tex]

(3) [tex] \nabla \cdot i_r [/tex]

(4) [tex] \nabla \times \nabla \cdot \theta [/tex]

...

The definition for [tex] \nabla [/tex] in cylindrical coordinate is:

[tex]\nabla f = \frac{\partial f}{\partial r} \hat r + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat \theta + \frac{\partial f}{\partial z} \hat z [/tex]

So...

(1) I first take [tex] \frac{\partial \theta}{\partial r} [/tex] which equals 0. Next I take [tex] \frac{1}{r}\frac{\partial \theta}{\partial \theta} [/tex] which equals [tex]\frac{1}{r}\hat \theta [/tex], etc...

so for problem (1)

[tex] \nabla \theta = \frac{1}{r}\hat \theta [/tex]

(2)

[tex] \nabla r^4 = 4r^3 \hat r [/tex]

(3)

[tex] \nabla \cdot i_r = \frac{1}{r} [/tex] from the definition of [tex] div [/tex] in cylindrical coordinates:

[tex] \nabla \cdot \vec F = \frac{1}{r}\frac{\partial}{\partial r} (r F_r) + ... [/tex]

(4)

[tex] \nabla \cross \nabla \theta [/tex]

Well from (1) I have [tex] \nabla \theta = \frac{1}{r}\hat \theta [/tex]

The general definition of curl is:

[tex] \nabla \times \vec F = \frac{1}{abc} \left| \begin{array}{ccc} a\hat i_1 & b\hat i_2 & c\hat i_3 \\ \partial u & \partial v & \partial w \\ aF_1 & bF_2 & cF_3 \end{array} \right| [/tex]

where: a=1, b=r, c=1, u=r, v=theta, w=z

Then plugging in the values I get:

[tex] \nabla \times \frac{1}{r}\hat \theta = \frac{1}{r} \left| \begin{array}{ccc} \hat r & r\hat \theta & \hat z \\ \partial r & \partial \theta & \partial z \\ 0 & r(\frac{1}{r}) & 0 \end{array} \right| = 0 [/tex] which is equal to 0, as it should...

So does everything look sound? Is my thought process here ok? I\d appreciate any help.

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# Homework Help: Cylindrical Coordinates -Div/Curl/Grad

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