# Convert Derivative in Polar to Cartesian

• mysubs
In summary, the conversation is about converting from cylindrical coordinates to cartesian coordinates and using the chain rule to find the equivalent expression. The question is whether using the formula dr=(xdx+ydy)/(x2+y2)1/2 is correct, and if not, how to properly work with it. The response confirms that this formula is correct and provides the partial derivative equation for x.
mysubs
Hello,

I want to convert from cylindrical (r,a,z) --to--> cartesian (x,y,z). However, I'm a not very confident about my level at this.

Say, I have dm/dr, a derivative in polar, and I want to find the equivalent expression in cartesian. (d: is partial derivative here).

I thought about using chain rule, but [r] depends on both [x] and [y]. Is doing it this way: dr=(xdx+ydy)/(x2+y2)1/2, correct? and If not why?

How should I work with this?

Yes, that's the way.
So

$$\frac{\partial m}{\partial x} = \frac{\partial m}{\partial r} \frac{\partial r}{\partial x}$$
where
$$\frac{\partial r}{\partial x} = \frac{x}{\sqrt{x^2 + y^2}}$$

## 1. What is the formula for converting a derivative in polar coordinates to cartesian coordinates?

The formula for converting a derivative from polar coordinates to cartesian coordinates is:
dy/dx = (dy/dθ * cosθ) - (y * sinθ) / (dx/dθ * cosθ) - (x * sinθ)

## 2. How do you convert a polar derivative to cartesian coordinates graphically?

To convert a polar derivative to cartesian coordinates graphically, you can plot the polar coordinates on a polar graph and then use the formula mentioned above to find the corresponding cartesian coordinates. Alternatively, you can also use a conversion tool or calculator to get the cartesian coordinates.

## 3. What does the term "derivative" mean in the context of polar coordinates?

In the context of polar coordinates, the term "derivative" refers to the rate of change of a function with respect to the angle, θ. It represents the slope of the tangent line to the polar curve at a given point.

## 4. Can you convert a cartesian derivative to polar coordinates?

Yes, it is possible to convert a cartesian derivative to polar coordinates. The formula for converting a cartesian derivative to polar coordinates is:
dy/dθ = (dy/dx * cosθ) + (y * sinθ) * (dx/dx * cosθ) - (x * sinθ)

## 5. Why is it useful to convert a derivative from polar to cartesian coordinates?

Converting a derivative from polar to cartesian coordinates allows us to analyze and graph polar functions in a familiar cartesian coordinate system. It also helps in finding the slope of a curve at a particular point and determining the rate of change of a function in different directions.

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