Mapping the Square and Circle Regions: Finding Ratios

[atomicpedals]

I've been working at this problem off and on all day (I come back to it after finishing others hoping I'll get a great idea)... I still have no clue how I should even start. Any hints in the right direction would be most welcome. So, without further ado, here's the question:


Show that the square R(xy):0.9 $$\leq$$ x $$\leq$$ 1.1, -0.1$$\leq$$ y $$\leq$$ 0.1 corresponds to the region R(uv) bounded by arcs of the circles u^2+v^2=e^1.8, u^2+v^2=e^2.2 and the rays v= +/-(tan 0.1)u, u$$\geq$$ 0, and find the ratio of the area of R(uv) to that of R(xy).


If I'm reading the problem correctly it's essentially a mapping. And I'm pretty sure that a Jacobian determinant will be involved in some way. Do I construct a Jacobian from the forms of R(xy) and R(uv)? But how does that get me the requested ratio?

 

[mighty2000]

it's important to approach problems in a systematic and logical manner. In this case, it seems like you have identified the key concepts that are involved in solving this problem - mapping and Jacobian determinants. These are both important tools in understanding and visualizing geometric transformations.

To start, I would suggest breaking down the problem into smaller, more manageable parts. First, try to understand the given region R(xy). What does it look like? What are its boundaries? How can you represent it mathematically? By answering these questions, you can gain a better understanding of the given region and its properties.

Next, focus on the region R(uv) and try to visualize how it relates to R(xy). Are there any similarities or differences between the two regions? This will help you determine the correct mapping function between the two regions.

Once you have a mapping function, you can use the Jacobian determinant to calculate the ratio of the areas between the two regions. Remember that the Jacobian determinant represents the change in area under a transformation.

Another helpful tip is to draw diagrams or use visual aids to help you better understand the problem. This can also help you identify any patterns or relationships between the two regions.

In summary, take a step back and break down the problem into smaller parts. Use your understanding of mapping and Jacobian determinants to find a suitable transformation between the two regions. And don't be afraid to draw diagrams or use visual aids to help you better understand the problem. Good luck!