In one line
If \(u(x,y,z)\) is homogeneous of degree \(n\), then \(xu_x+yu_y+zu_z=nu\); it is the same theorem with one more term.
Explained simply
Think of a box with sides \(x,y,z\). Its volume \(V=xyz\) has degree 3: doubling every side makes it 8 times bigger. Euler's theorem with three variables says \(xV_x+yV_y+zV_z=x(yz)+y(xz)+z(xy)=3xyz=3V\). Each variable gets its own weighted term, and the total is the degree times the function.
Technically, \(u\) is homogeneous of degree \(n\) if \(u(tx,ty,tz)=t^nu(x,y,z)\). Then \(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}+z\dfrac{\partial u}{\partial z}=nu\). This is a self-learning topic. The same deductions hold: if \(f(u)\) is homogeneous of degree \(n\), then \(xu_x+yu_y+zu_z=n\dfrac{f(u)}{f'(u)}\).
Why it matters: in physics, the gravitational and electric potentials \(\dfrac{k}{r}\) with \(r=\sqrt{x^2+y^2+z^2}\) are homogeneous of degree \(-1\), and Euler's theorem gives quick relations between the potential and its rate of change along the radius. In thermodynamics and economics, quantities that scale with size are homogeneous of degree 1 and obey the same relation.
How it works
- Check homogeneity with the \(t\)-test: replace \(x,y,z\) by \(tx,ty,tz\).
- Find the degree \(n\).
- Write \(xu_x+yu_y+zu_z=nu\) directly.
- If the outer function spoils homogeneity, use \(f(u)\) and \(G(u)=n\dfrac{f(u)}{f'(u)}\).
Key formula
\(xu_x+yu_y+zu_z=nu\), where \(n\) is the degree of \(u(x,y,z)\).
Quick example
Let \(u=\dfrac{1}{\sqrt{x^2+y^2+z^2}}\), the inverse distance from the origin (used for gravitational and electric potentials). Degree \(=-1\). So Euler gives \(xu_x+yu_y+zu_z=-u\).
Direct check: \(u_x=-x(x^2+y^2+z^2)^{-3/2}\), and similarly for \(y,z\). Then \(xu_x+yu_y+zu_z=-(x^2+y^2+z^2)(x^2+y^2+z^2)^{-3/2}=-u\). Correct.
Another: \(u=\log\dfrac{x^4+y^4+z^4}{x+y+z}\). Here \(e^u=\dfrac{x^4+y^4+z^4}{x+y+z}\) has degree 3, so \(xu_x+yu_y+zu_z=3\dfrac{e^u}{e^u}=3\).
Common mistakes
- Forgetting the \(zu_z\) term.
- Treating \(\log(\dots)\) as homogeneous; use \(e^u\) instead.
- Getting the degree of a root wrong: \(\sqrt{x^2+y^2+z^2}\) has degree 1, so its reciprocal has degree \(-1\).
How MU asks it
Being self-learning, it is usually a short 3–5-mark part: "If \(u=x^3+y^3+z^3+3xyz\), verify Euler's theorem" or "If \(u=\sin^{-1}\dfrac{x+y+z}{\sqrt x+\sqrt y+\sqrt z}\), find \(xu_x+yu_y+zu_z\)" (answer \(\dfrac12\tan u\)). Show the \(t\)-test, the degree, and quote the theorem before substituting.