Convert Cartesian To Cylindrical Limits

[farso]

Homework Statement

Hi

I am converting from Cartesian co-ordinates to cylindrical co-ordinates systems and can do the conversion fine, but am unsure about how to convert the limits.

Q)

Region V is given in Cartesian by: F(x,y,z) = xi+yj+z(x^{2}+y^{2})k

Where x^{2}+y^{2} \leq 1
and 0 \leq z \leq 2

Homework Equations

x = \rho cos(\varphi), y = \rho sin(\varphi), z = z
cos^{2}(x)+sin^{2}(x) = 1

The Attempt at a Solution

F(\rho,\varphi,z) = \rho cos(\varphi)\widehat{\rho} + \rho sin(\varphi)\widehat{\varphi} + z\rho^{2}\widehat{z}

I know that z has stayed the same as it has not changed, but I am not sure how to do the \rho and \varphi.

If you could help me on how to do this id really appreciate it.

Many thanks in advance

 

[gabbagabbahey]

F(\rho,\varphi,z) = \rho cos(\varphi)\widehat{\rho} + \rho sin(\varphi)\widehat{\varphi} + z\rho^{2}\widehat{z}

It looks as though you've magically replaced \hat{\mathbf{x}} and \hat{\mathbf{y}} with \hat{\mathbf{\rho}}[/itex] and \hat{\mathbf{\varphi}}[/itex]...are you really allowed to do that?<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" /><br /> <br /> What are \hat{\mathbf{x}} and \hat{\mathbf{y}} in terms of \hat{\mathbf{\rho}}[/itex] and \hat{\mathbf{\varphi}}[/itex]?<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I know that z has stayed the same as it has not changed, but I am not sure how to do the \rho and \varphi. </div> </div> </blockquote><br /> Well, your other boundary is x^2+y^2\leq 1...so, what is x^2+y^2 is cylindrical coordinates?

 

[farso]

It looks as though you've magically replaced \hat{\mathbf{x}} and \hat{\mathbf{y}} with \hat{\mathbf{\rho}}[/itex] and \hat{\mathbf{\varphi}}[/itex]...are you really allowed to do that?<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" /><br /> <br /> What are \hat{\mathbf{x}} and \hat{\mathbf{y}} in terms of \hat{\mathbf{\rho}}[/itex] and \hat{\mathbf{\varphi}}[/itex]?<br />

<br /> <br /> Sorry, I am unsure what you mean here? <br /> <br /> <blockquote data-attributes="" data-quote="gabbagabbahey" data-source="post: 2695089" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> gabbagabbahey said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Well, your other boundary is x^2+y^2\leq 1...so, what is x^2+y^2 is cylindrical coordinates? </div> </div> </blockquote><br /> Ok, I think I see what you mean here... Would it be:<br /> <br /> \rho \leq 1<br /> x^{2}+y^{2} \leq 1<br /> = \rho cos^{2}(\varphi) + \rho sin^{2}(\varphi)<br /> <br /> If this is correct, I am still unsure how to get \varphi?<br /> <br /> Thanks for the speedy response!

 

[gabbagabbahey]

Sorry, I am unsure what you mean here?

When you substitute x=\rho\cos\varphi and y=\rho\sin\varphi into \textbf{F}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z(x^2+y^2)\hat{\mathbf{z}}, you get \textbf{F}=\rho\cos\varphi\hat{\mathbf{x}}+\rho\sin\varphi\hat{\mathbf{y}}+z\rho^2\hat{\mathbf{z}}, not \textbf{F}=\rho\cos\varphi\hat{\mathbf{\rho}}+\rho\sin\varphi\hat{\mathbf{\varphi}}+z\rho^2\hat{\mathbf{z}}

\hat{\mathbf{x}}\neq\hat{\mathbf{\rho}} \;\;\;\;\;\;\text{and}\;\;\;\;\;\;\; \hat{\mathbf{y}}\neq\hat{\mathbf{\varphi}}[/itex]<br /> <br /> In your text, you should have derived expressions of the form \hat{\mathbf{x}}=f(\rho,\theta)\hat{\mathbf{\rho}}+g(\rho\theta)\hat{\mathbf{\varphi}} and \hat{\mathbf{y}}=F(\rho,\theta)\hat{\mathbf{\rho}}+G(\rho\theta)\hat{\mathbf{\varphi}} ...use those<br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Ok, I think I see what you mean here... Would it be:<br /> <br /> \rho \leq 1<br /> x^{2}+y^{2} \leq 1<br /> = \rho cos^{2}(\varphi) + \rho sin^{2}(\varphi)<br /> <br /> If this is correct, I am still unsure how to get \varphi?<br /> <br /> Thanks for the speedy response! </div> </div> </blockquote><br /> Well, you know that 0\leq\rho\leq 1, so these are your integration limits for \rho. As for \varphi, there are no further restrictions on its value, so you just go with the conventional 0\leq\varphi\leq 2\pi.

 

[farso]

When you substitute x=\rho\cos\varphi and y=\rho\sin\varphi into \textbf{F}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z(x^2+y^2)\hat{\mathbf{z}}, you get \textbf{F}=\rho\cos\varphi\hat{\mathbf{x}}+\rho\sin\varphi\hat{\mathbf{y}}+z\rho^2\hat{\mathbf{z}}, not \textbf{F}=\rho\cos\varphi\hat{\mathbf{\rho}}+\rho\sin\varphi\hat{\mathbf{\varphi}}+z\rho^2\hat{\mathbf{z}}

\hat{\mathbf{x}}\neq\hat{\mathbf{\rho}} \;\;\;\;\;\;\text{and}\;\;\;\;\;\;\; \hat{\mathbf{y}}\neq\hat{\mathbf{\varphi}}[/itex]<br /> <br /> In your text, you should have derived expressions of the form \hat{\mathbf{x}}=f(\rho,\theta)\hat{\mathbf{\rho}}+g(\rho\theta)\hat{\mathbf{\varphi}} and \hat{\mathbf{y}}=F(\rho,\theta)\hat{\mathbf{\rho}}+G(\rho\theta)\hat{\mathbf{\varphi}} ...use those

I have searched my notes, and my textbooks, but do not have anything which seems to cover this. Is there a particular name for this operation that I may be able to search up and check out some examples?<br /> <br /> Thanks

 

[gabbagabbahey]

I have a hard time believing that your course textbook doesn't cover unit vector conversions. What text are you using for this course?

Anyways, see here

 

[farso]

I have a hard time believing that your course textbook doesn't cover unit vector conversions. What text are you using for this course?

Anyways, see here

Thats amazing, thanks. I hope I have now got it correct.

So...
F(\rho,\varphi,z) = \rho cos(\varphi)\widehat{i} + \rho sin(\varphi)\widehat{j} + z\rho^{2}\widehat{z}

and

\widehat{\rho} = \widehat{i} cos(\varphi) + \widehat{j} sin(\varphi)

Subbing that in I am just left with:
F(\rho,\varphi,z) = \rho \widehat{\rho} + z\rho^{2}\widehat{z}

Is that looking to be correct?

Thanks for all your help, you have been very good to me.

BTW, I am using KA Stroud, Advanced Engineering Maths, but cannot find it in there still.

 

[gabbagabbahey]

Looks good to me!