• stevendaryl
In summary, the conversation discusses a two-dimensional coordinate system and a new coordinate system with a function theta(u,v) that relates the partial derivatives of x and y. The question is whether these four equations imply that theta is constant, indicating a linear relationship between the coordinate systems. It is concluded that this is the case, as computing the cross derivatives results in equations that require theta to have partial derivatives of zero.
stevendaryl
Staff Emeritus
Suppose that I have a two-dimensional coordinate system $(x,y)$ and I change to a new coordinate system $(u,v)$. What I know is that there is some function $\theta(u,v)$ such that:
1. $\dfrac{\partial x}{\partial u} = cos(\theta)$
2. $\dfrac{\partial x}{\partial v} = -sin(\theta)$
3. $\dfrac{\partial y}{\partial u} = sin(\theta)$
4. $\dfrac{\partial y}{\partial v} = cos(\theta)$
My question is: do these 4 equations imply that $\theta =$ constant? (so that the relationship between the coordinate systems is linear)

Looks like it yes, if you compute the cross derivatives
## \frac{\partial^2x}{\partial u\partial v} =-\sin\theta\frac{\partial\theta}{\partial v}=-\cos\theta\frac{\partial\theta}{\partial u}\\\frac{\partial^2y}{\partial u\partial v} =\cos\theta\frac{\partial\theta}{\partial v}=-\sin\theta\frac{\partial\theta}{\partial u} ##
you get a set of equations that require ## \frac{\partial\theta}{ \partial u}=\frac{\partial\theta}{ \partial v}=0##

wabbit said:
Looks like it yes, if you compute the cross derivatives
## \frac{\partial^2x}{\partial u\partial v} =-\sin\theta\frac{\partial\theta}{\partial v}=-\cos\theta\frac{\partial\theta}{\partial u}\\\frac{\partial^2y}{\partial u\partial v} =\cos\theta\frac{\partial\theta}{\partial v}=-\sin\theta\frac{\partial\theta}{\partial u} ##
you get a set of equations that require ## \frac{\partial\theta}{ \partial u}=\frac{\partial\theta}{ \partial v}=0##

Thank you! I tried exactly that, but I made a stupid sign error, and found them consistent.

Ah yes, signs are the spawn of the devil :)

## 1. What is a coordinate change?

A coordinate change is the process of converting coordinates from one system to another. It involves transforming the position of a point or object from one set of coordinates to another using mathematical equations.

## 2. Why is coordinate change important in science?

Coordinate change is important in science because it allows us to accurately describe and measure the position of objects and phenomena in the natural world. It also helps us to analyze data and make predictions using mathematical models.

## 3. What are the different types of coordinate systems used in science?

The most commonly used coordinate systems in science are the Cartesian coordinate system, polar coordinate system, and spherical coordinate system. These systems have different ways of defining and measuring coordinates, but they all serve the same purpose of locating points in space.

## 4. How do you perform a coordinate change?

The specific method for performing a coordinate change depends on the type of coordinate system being used. In general, it involves using mathematical equations or transformations to convert coordinates from one system to another. This can be done manually or with the help of computer software.

## 5. Can coordinate change affect the accuracy of measurements?

Yes, coordinate change can affect the accuracy of measurements if it is not performed correctly. Errors in the conversion process or imprecise equations can lead to inaccuracies in the final coordinates. It is important to use reliable methods and double-check calculations to ensure the accuracy of measurements.

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