# Change of variables/ Transformations part 2

• MHB
• Joe20
In summary, when setting u and v expressions into the u-v plane for a question, you can start by substituting x = 2u and y = 3v, and then converting to polar coordinates to find the parametric equations for a circle or ellipse. For a circle with radius r, the equations are x = r cos(t) and y = r sin(t), and for an ellipse with axes a and b, the equations are x = a cos(t) and y = b sin(t).
Joe20
I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?

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Alexis87 said:
I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?
You could start by letting $x = 2u$ and $y = 3v$. You might then want to make a further change, from cartesian to polar coordinates.

You probably know that parametric equations for a circle with radius r, centered at (0, 0), x^2+ y^2= r^2, are x= r cos(t), y= r sin(t) because x^2+ y^2= r^2cos^2(t)+ r^2 sin^2(t)= r^2(cos^2(t)+ sin^2(t))= r^2.

It should not be too much of a "jump" to see that parametric equations for the ellipse, with axes of length a and b in the x and y direction, respectively, x^2/a^2+ y^2/b^2= 1, are x= a cos(t), y= b sin(t).

## 1. What is the purpose of a change of variables/ transformations in science?

A change of variables/ transformations is used to make complex equations or problems easier to solve by substituting new variables in place of the original ones. This can also be used to transform the shape or orientation of a graph or data set.

## 2. How do you determine the appropriate change of variables for a given problem?

The appropriate change of variables can be determined by analyzing the given problem and identifying any patterns or relationships between the variables. It is also helpful to consider the desired outcome and how the change of variables can simplify the problem.

## 3. What are the most commonly used types of transformations in science?

The most commonly used types of transformations in science include linear transformations, logarithmic transformations, and exponential transformations. Other types such as trigonometric transformations or power transformations may also be used depending on the problem at hand.

## 4. How does a change of variables affect the interpretation of data?

A change of variables can significantly impact the interpretation of data by altering the shape or distribution of the data. This can reveal hidden patterns or relationships that were not apparent before the transformation. It can also make data more easily understandable and comparable to other data sets.

## 5. Can a change of variables be applied to any mathematical problem?

A change of variables can be applied to many mathematical problems, but it may not always be the most effective or appropriate approach. It is important to carefully consider the problem and the desired outcome before deciding to use a change of variables. In some cases, other methods such as integration or differentiation may be more suitable.

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