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Eliminate Cross-Term by Change in Variables

elimxy3.html

Exploration

Show that applying the change in variables x'=k x+y and y'=k y-x, where

"elimxy3_1.gif",

to the equation "elimxy3_2.gif" is equivalent to rotating the quadratic by an angle θ given by

"elimxy3_3.gif"

and scaling the quadratic by a scale factor

"elimxy3_4.gif".

Approach

Create a quadratic and rotate and scale it as specified. Compare the result to the result of elimxy2.html.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create a quadratic.

Clear[a,b,c,d,e,f];
Q1=Quadratic2D[a,b,c,d,e,f];

Rotate it by the specified angle.

Clear[k];
Q2=Rotate2D[Q1,ArcTan[1/k]] //Simplify

"elimxy3_5.gif"

As shown in elimxy2.html, the x y term must vanish.

Q2[[2]]=0;Q2

"elimxy3_6.gif"

Scale as specified.

Q3=Scale2D[Q2,1/Sqrt[1+k^2]] //Simplify

"elimxy3_7.gif"

Simplify, showing the same result as elimxy2.html.

Q4=Q3 /. {Sqrt[1+k^(-2)]*k->Sqrt[1+k^2]} //Simplify

"elimxy3_8.gif"


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