Simple integral in cylindrical coordinates

Click For Summary
SUMMARY

The discussion centers on calculating the integral \(\oint[\hat{r}+\hat{z}]d\phi\) in cylindrical coordinates. The user initially computes this integral as \(2\pi[\hat{r}+\hat{z}]\) but later converts it to Cartesian coordinates, yielding \(2\pi\hat{z}\). The discrepancy arises from the fact that \(\hat{r}\) is a function of \(\phi\), which the user fails to account for in the cylindrical coordinate system. Understanding the relationship between \(\hat{r}\) and \(\phi\) is crucial for accurate calculations in cylindrical coordinates.

PREREQUISITES
  • Understanding of cylindrical coordinates and their components (\(\hat{r}, \hat{\phi}, \hat{z}\))
  • Knowledge of vector calculus and line integrals
  • Familiarity with coordinate transformations between cylindrical and Cartesian systems
  • Basic understanding of integrals in multi-variable calculus
NEXT STEPS
  • Study the properties of cylindrical coordinates and their applications in vector calculus
  • Learn about the relationship between \(\hat{r}\) and \(\phi\) in cylindrical coordinates
  • Explore line integrals in different coordinate systems, focusing on cylindrical coordinates
  • Review examples of converting integrals from cylindrical to Cartesian coordinates
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with integrals in cylindrical coordinates, particularly those seeking to deepen their understanding of vector calculus and coordinate transformations.

Vagrant
Messages
195
Reaction score
1

Homework Statement


As a part of bigger HW problem, I need to calculate the integral:
[itex]\oint[/itex][[itex]\hat{r}[/itex]+[itex]\hat{z}[/itex]]d[itex]\phi[/itex]

Homework Equations





The Attempt at a Solution



In cylindrical coordinates:
=[[itex]\hat{r}[/itex]+[itex]\hat{z}[/itex]] [itex]\oint[/itex]d[itex]\phi[/itex]
=2∏[[itex]\hat{r}[/itex]+[itex]\hat{z}[/itex]]

On the other hand if I convert it to Cartesian coordinates:
[itex]\oint[/itex][cos[itex]\phi[/itex][itex]\hat{x}[/itex]+sin[itex]\phi[/itex][itex]\hat{y}[/itex]+ [itex]\hat{z}[/itex]]d[itex]\phi[/itex]
=2∏[itex]\hat{z}[/itex]

So, what is it that I am doing wrong in the case with cylindrical coordinates? I'm sure I'm missing something very basic.
Thanks.
 
Physics news on Phys.org
Vagrant said:

Homework Statement


As a part of bigger HW problem, I need to calculate the integral:
[itex]\oint[/itex][[itex]\hat{r}[/itex]+[itex]\hat{z}[/itex]]d[itex]\phi[/itex]

Homework Equations





The Attempt at a Solution



In cylindrical coordinates:
=[[itex]\hat{r}[/itex]+[itex]\hat{z}[/itex]] [itex]\oint[/itex]d[itex]\phi[/itex]
=2∏[[itex]\hat{r}[/itex]+[itex]\hat{z}[/itex]]

On the other hand if I convert it to Cartesian coordinates:
[itex]\oint[/itex][cos[itex]\phi[/itex][itex]\hat{x}[/itex]+sin[itex]\phi[/itex][itex]\hat{y}[/itex]+ [itex]\hat{z}[/itex]]d[itex]\phi[/itex]
=2∏[itex]\hat{z}[/itex]

So, what is it that I am doing wrong in the case with cylindrical coordinates? I'm sure I'm missing something very basic.
Thanks.
[itex]\hat{r}[/itex] is a function of [itex]\phi\ .[/itex]
 
Sorry, but could you please explain this a bit?
I can visualize that [itex]\hat{r}[/itex] is a function of [itex]\phi[/itex](unit circle on x-y plane centered at origin) for cartesian coordinates, but I'm drawing a blank when it comes to cylindrical coordinates.
 

Similar threads

Surface Area of a Sphere in Spherical Coordinates
  • · Replies 14 ·
Replies
14
Views
3K
The relation between the normal and the slope of a cylindrical curve
  • · Replies 2 ·
Replies
2
Views
2K
Stokes' Theorem 'corollary' integral in cylindrical polar coordinates
  • · Replies 6 ·
Replies
6
Views
2K
Vector Field Transformation to Spherical Coordinates
  • · Replies 5 ·
Replies
5
Views
2K
Surface Integral of a sphere
  • · Replies 9 ·
Replies
9
Views
2K
Vector field in cylindrical coordinates
  • · Replies 2 ·
Replies
2
Views
3K
Flux in a rotated cylindrical coordinate system
  • · Replies 7 ·
Replies
7
Views
3K
What's the integral of a unit vector?
  • · Replies 9 ·
Replies
9
Views
7K
Surface element in cylindrical coordinates
  • · Replies 1 ·
Replies
1
Views
4K
How to set bounds in cylindrical coordinates analytically?
  • · Replies 4 ·
Replies
4
Views
2K