Euler's theorems

Euler's theorems

1. Homogeneous Function

$f(x,y)$ is homogeneous of degree $n$ if, for all $t>0$:

Equivalently, every term has same total degree in $x,y$ after factoring. Examples:

• $f= x^2+y^2-xy$ : degree $2$ ($t^2$ factor)
• $f= sin^{-1}(y/x)$ : degree $0$ (ratio cancels $t$)
• $f= (x^3+y^3)/(x+y)$ : degree $2$
• $f= x log(y/x)$ : degree $1$ ($t$ factor: $tx log(ty/tx)= t·x log(y/x)$)
  Check: replace $x→tx, y→ty$, pull $t^n$ out.

2. Euler's Theorem — First Order

If $f(x,y)$ is homogeneous of degree $n$ and has continuous first partials, then:

For three variables $f(x,y,z)$ homogeneous degree $n$: $x f_x + y f_y + z f_z = n f$.

Proof sketch: differentiate $f(tx,ty)=t^n f(x,y)$ w.r.t. $t$ at $t=1$ using chain rule.

3. Euler's Theorem — Second Order (frequently asked)

If $f$ homogeneous degree $n$:

Derived by applying Euler to $f_x, f_y$ (each homogeneous degree $n-1$) and combining.

More generally, with $u = f(x,y)$ viewed as $u(x,y)$, examiners may ask for $x ∂u/∂x + y ∂u/∂y$ via Euler even when $u$ is not directly homogeneous but $z = F(u)$ is — then apply chain rule.

4. Trick for $z = F(u)$ Where $u$ Is Homogeneous

If $u = f(x,y)$ homogeneous degree $n$ and $z = Φ(u)$ (e.g., $z = sin u$), then:

Example: $u = (x^2+y^2)/(x+y)$ (degree 1), $z = log u$ → $x z_x + y z_y = 1·u·(1/u)=1$? Wait compute: $Φ'(u)=1/u$, so $n u·1/u = n =1$ ✓

General formula: $x z_x + y z_y = n·u·Φ'(u)$.

5. Worked Examples

(a) f= x^3+y^3-3x y^2 : degree 3 ⇒ x f_x + y f_y =3f
    Check: f_x=3x^2-3y^2, f_y=3y^2-6xy ⇒ x f_x+y f_y=3x^3-3x y^2+3y^3-6x y^2=3f ✓

(b) f= sin^{-1}(y/x): f(tx,ty)=sin^{-1}(y/x)=f ⇒ n=0 ⇒ x f_x + y f_y =0
    Compute: f_x= -y/(x√(x^2-y^2)), f_y=1/√(x^2-y^2) ⇒ x f_x+y f_y=0 ✓

(c) u= (x^2+y^2)^{1/2} (degree 1): e.g., show x^2 u_{xx}+2xy u_{xy}+y^2 u_{yy}=0 since n(n-1)=0 ✓

(d) z = tan^{-1}(y/x) (degree 0): x z_x + y z_y =0 ⇒ z_x= -y/(x^2+y^2), z_y= x/(x^2+y^2) ⇒ sum 0 ✓

(e) f= x^n log(y/x): check homogeneous degree n? f(tx,ty)= t^n x^n [log y/x] = t^n f ⇒ degree n ⇒ x f_x+y f_y=n f ✓

(f) Second-order: f= x^4+y^4 degree 4 ⇒ x^2 f_{xx}+2xy f_{xy}+y^2 f_{yy}=12 f

6. Exam Points

• Test homogeneity and find degree $n$ — 3 marks, very frequent.
• State and prove Euler first-order $x f_x + y f_y = n f$ — 5-7 marks, almost certain.
• Verify Euler for given $f$ — 5 marks, very frequent.
• Second-order form — 5 marks, frequent.
• Application where $z = F(u)$ with $u$ homogeneous — 7 marks, very frequent (e.g., $z = sin^{-1}(x/y) + cos^{-1}(y/x)$ type).