A multivariate function, $f(\vec{x})$, is said to be homogeneous of order $n$, if $$f(\lambda\vec{x}) = \lambda^n f(\vec{x})$$
One of the most interesting and relevant properties of homogeneous functions is Euler’s theorem.
Euler’s Theorem for Homegeneous Function
Differentiating the definition by $\lambda$ one gets: $$n\lambda^{n-1}f(\vec{x}) = \sum_j\frac{\partial f(\lambda\vec{x})}{\partial(\lambda x_j)}\frac{d(\lambda x_j)}{d\lambda}$$ $$=\vec\nabla f(\lambda\vec{x}) \cdot \vec{x}$$ Putting $\lambda = 1$, we get Euler’s theorem for homogeneous functions, $$nf(\vec{x}) = \vec\nabla f(\vec{x})\cdot\vec{x},$$ which relates the value of the function at a points to the values of all the gradient of the function and the point itself.
Consequence for Entropy and Energy
Given that entropy and energy are both first-order homogeneous functions, we can now write: $$S(U, V, N_j) = \frac{\partial S}{\partial U} U + \frac{\partial S}{\partial V} V + \sum_j \frac{\partial S}{\partial N_j} N_j$$ $$U(S, V, N_j) = \frac{\partial U}{\partial S} S + \frac{\partial U}{\partial V} V + \sum_j \frac{\partial U}{\partial N_j} N_j$$
Now, what are these partial derivatives? We will talk more about them in next.