Converting Cartesian to Cylindrical Coordinates

[Shaheers]

Homework Statement

Hi i have a question that how to convert the co-ordinates according to following statement?
Its no difficult to solve the simple conversion but the bold one are confusing me.


A=4x-2y-4z
Transform it into cylindrical co-ordinate system at P (ρ=4, ϕ=120o ,z=2)

B=15x+10y
Transform it into cylindrical co-ordinate system at P(x=3, y=4, z=-1)





2. The attempt at a solution

Equations for converting Cartesian to cylindrical coordinates:

r= [x² + y^21/2
ϕ= tan^-1= y/x
z=z

by putting values
For First question:
r=√20=4.472
ϕ= -26.57
z=-4

For Second question:
r= 18.03
ϕ= 33.69
z=0

 

[vela]

This is common point of confusion. What you've done is convert the Cartesian coordinates (4, -2, -4) to the equivalent cylindrical coordinates. What the problem is asking you to do, however, is different.

Take a look at the diagram at the start of this page:

http://mathworld.wolfram.com/CylindricalCoordinates.html

At each point, there are associated unit vectors $\hat{\rho}$, $\hat{\phi}$, and $\hat{z}$. Their directions depend on which point you're at. For example, if you're at the point (1, 0, 0), $\hat{\rho}$ would point along the +x-direction, and $\hat{\phi}$ would point in the +y direction. If you're at the point (0, 1, 0), $\hat{\rho}$ would point in the +y direction, and $\hat{\phi}$ would point in the -x direction.

Given the information in bold, you can figure out what $\hat{\rho}$ and $\hat{\phi}$ are. $\hat{z}$ always points in the +z direction. Once you know what the unit vectors are equal to, you're supposed to express the vectors A and B in terms of them.

 

[Shaheers]

unfortunately i can't understand from that given link, kindly give an example by solving one of the equation or any of. . .

 

[vela]

Sorry, we don't do your homework for you here. If you can't see what I tried to explain from the diagram, I suggest you consult your textbook for a more detailed explanation.

 

[Shaheers]

I doesn't mean that, i just want to say that, if you please tell this by example by taking any random point P, if such an example is given on textbook, i don't need to refer from internet

 

[vela]

My point was that I would have liked to see even a minimal attempt from you to try understand what's going on instead of simply saying "I don't get it."

At each point, there are associated unit vectors $\hat{\rho}$, $\hat{\phi}$, and $\hat{z}$. Their directions depend on which point you're at. For example, if you're at the point (1, 0, 0), $\hat{\rho}$ would point along the +x-direction, and $\hat{\phi}$ would point in the +y direction. If you're at the point (0, 1, 0), $\hat{\rho}$ would point in the +y direction, and $\hat{\phi}$ would point in the -x direction.

Say you have a vector field $\vec{V}(\vec{r})$, and at both $\vec{r}=(1, 0, 0)$ and $\vec{r}=(0, 1, 0)$, it points in the +x direction.

At the point (1, 0, 0), you'd say $\vec{V}(1, 0, 0) = \hat{\rho}$. At the point (0, 1, 0), however, the basis vectors point in different directions. In this case, you'd have $\vec{V}(0, 1, 0) = -\hat{\phi}$.