How do I express an equation in Polar coordinates as a Cartesian one.

In summary, the conversation discusses converting a polar function, represented by the equation $$ \psi = P(\theta)R(r) $$, into cartesian coordinates. This is done by using the relations $$ x = r Cos(\theta) $$, $$ y = r Sin(\theta) $$, $$ r = \sqrt{x^{2}+y^{2}} $$, and $$ \theta = arcTan(\frac{y}{x}) $$. However, since the functions R and P have first and second derivatives dependent on x and y, it is not possible to directly replace the relations. Instead, the Laplacian of psi must be expressed in cartesian coordinates by setting up expressions such as $$ \frac{\partial P(\theta(x
  • #1
JorgeM
30
6
TL;DR Summary
I need to convert an equation that is on polar coordinates into a cartesian but as soon I start doing that I got confused and I'm not really shure about what to do.
I got a polar function.

$$ \psi = P(\theta )R(r) $$

When I calculate the Laplacian:
$$ \ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}}
$$

Now I need to convert this one into cartesian coordinates and then it results very difficult to me because I know how to convert simple equations using the simple relations:$$ x = r Cos(\theta ) $$$$ y = r Sin(\theta ) $$

$$ r= \sqrt{x^{2}+y^{2}} $$

$$ \theta= arcTan( \frac{y}{x}) $$I can't figure out how to use this relations in order to replace them in my functions R and P since there are first and second derivatives of a functions dependent on x and y, so I can not just replace the relations (At least not directly).

What I need to do is to express the Laplacian of psi in cartesian coordinates.
Is there a way just to replace in them as :

$$ P(\theta) = F(x,y)$$
$$ P^{\prime}(\theta) = G(x,y)$$

$$ P^{\prime\prime}(\theta) = H(x,y)$$

Or how should I try?

Thanks A lot for your help
 
Mathematics news on Phys.org
  • #2
If you don't know P and R, the best you can do is setting up something like ##\displaystyle \frac{\partial P(\theta(x,y))}{\partial \theta}## and then applying the chain rule. It will still look quite awkward.
 

Related to How do I express an equation in Polar coordinates as a Cartesian one.

1. What is the difference between Polar and Cartesian coordinates?

Polar coordinates use a distance from the origin and an angle from the positive x-axis to locate a point, while Cartesian coordinates use a horizontal and vertical distance from the origin.

2. How do I convert an equation in Polar coordinates to Cartesian coordinates?

To convert an equation from Polar to Cartesian coordinates, you can use the following formulas:

  • x = r * cos(theta)
  • y = r * sin(theta)

where r is the distance from the origin and theta is the angle from the positive x-axis.

3. Can all equations in Polar coordinates be expressed in Cartesian coordinates?

Yes, all equations in Polar coordinates can be expressed in Cartesian coordinates using the conversion formulas mentioned above.

4. How do I graph an equation in Polar coordinates as a Cartesian one?

To graph an equation in Polar coordinates as a Cartesian one, you can plot points by converting them to Cartesian coordinates using the conversion formulas. Then, you can connect the points to create a graph.

5. Are there any special cases or exceptions when converting equations from Polar to Cartesian coordinates?

Yes, there are some special cases and exceptions when converting equations from Polar to Cartesian coordinates. For example, when the equation is in the form of r = a (a constant), it represents a circle with radius a in Cartesian coordinates. Also, when the equation is in the form of theta = b (a constant), it represents a vertical line passing through the origin in Cartesian coordinates.

Similar threads

Replies
13
Views
2K
  • General Math
Replies
7
Views
1K
  • General Math
Replies
4
Views
1K
Replies
2
Views
1K
Replies
1
Views
874
Replies
2
Views
1K
Replies
8
Views
451
  • General Math
Replies
7
Views
2K
  • Differential Geometry
Replies
9
Views
699
Back
Top