given the function f(x,y)=3x2+4xy+3y2 and u=x+y, v=x-y find the new function expressed in f(u,v). See description

Question

given the function f(x,y)=3x²+4xy+3y²

and u=x+y,

v=x-y.

find the new function expressed in f(u,v)

Answer 1

Since u = x+y and v = x-y, u+v = 2x. So, x = (u+v)/2.

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

So, 3x² + 4xy + 3y² = 3[(u+v)/2}² + 4[(u+v)/2]{(u-v)/2] + 3[(u-v)/2]² =

(3/4)(u² + 2uv + v²) + u² - v² + (3/4)(u² - 2uv + v²) =

(5/2)u² + (1/2)v²

Answer 2

Given: f(x,y)=3x²+4xy+3y², u=x+y, v=x-y.

Solution:

f(u,v)=f(x+y,x-y)= 2(x+y)²+4(x+y)(x-y)+3(x-y)² =

=2(x²+2xy+y²)+4(x² -y²)+3(x²+2xy+y²)=9x²+10xy+y²