Solve Change of Variable: Triangular to Square/Rectangle

In summary, the conversation discusses how to perform a change of variable on a triangular region to make it appear as a square or rectangular region, using the equations x = (1/2)(u+v) and y = (1/2)(u-v). It is noted that no continuous change of variable can make a 3-sided region into a 4-sided one. The conversation also mentions using u = x+y, v = x-y as coordinates to make two sides of the triangle correspond to u=0 and v=0, and the third side corresponds to u+v=2.
  • #1
Suitengu
46
0
[SOLVED] Change of Variable

In case of the triangular area with vertices: (-1 1) (0 0) and (1 1), how would you do a change of variable to make that look like a square or rectangular region in terms of u an v?

x = (1/2)(u+v) y = (1/2)(u-v)

so

u = x+y v = x-y
 
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  • #2
No (continuous) change of variable will make a 3 sided region into a 4 sided region!
 
  • #3
oh. well what i did was to do the change of variable and then do the intervals as if i had a whole quadrilateral region and then i halved the answer. It seems to be right but i am not sure however.
 
  • #4
Choosing u= x+y, v= x- y as coordinates will make two sides of the triangle, y= -x and y= x correspond to u= 0 and v= 0 respectively. The third side of the triangle, y= 1, becomes u+ v= 2.

I have no idea what you mean by "intervals" or what answer you "halved". Apparently this was part of a larger problem you haven't mentioned.
 

FAQ: Solve Change of Variable: Triangular to Square/Rectangle

What is a change of variable?

A change of variable is a mathematical technique used to simplify a problem by substituting one variable with another. This is especially useful in integration and optimization problems, where the new variable can make the problem easier to solve.

How do you solve a change of variable from triangular to square/rectangle?

To solve a change of variable from triangular to square/rectangle, you need to first plot the triangular region and the square/rectangle region on a graph. Then, choose a suitable variable substitution that will transform the triangular region into the square/rectangle region. Finally, rewrite the limits of integration using the new variable and solve the integral or optimization problem as usual.

What are some common variable substitutions for this type of problem?

Some common variable substitutions for a change of variable from triangular to square/rectangle include polar coordinates, trigonometric identities, and geometric transformations. The specific substitution to use will depend on the problem at hand.

What are the benefits of using a change of variable?

Using a change of variable can simplify a problem and make it easier to solve. It can also help to reveal patterns and relationships that may not have been apparent with the original variables. Additionally, it can be useful in transforming a problem into a more general form that can be applied to a wider range of scenarios.

Are there any limitations to using a change of variable?

While a change of variable can be a useful tool in solving problems, it may not always be applicable or may not always yield the desired results. This is especially true when dealing with non-linear functions or when the transformation results in overly complex expressions. It is important to carefully consider the suitability of a change of variable before implementing it in a problem.

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