Most undergraduate books on Thermodynamics offer intuitive definitions for intensive and extensive thermodynamic variables. Authors assert, for example, that the former is independent of the system's size while the latter is not. Other authors argue splitting the system into several subsystems and investigating if the quantity of interest changes or remains the same in those. While all approaches make physical sense, I haven't seen someone defining those two concepts in a rigorous manner (even now that I am in university).

In the question Understanding "natural variables" of the thermodynamic potentials using the example of the ideal gas, one person argued in the answers that an extensive quantity, say $A$ which is dependent from other thermodynamic variables $X_1, X_2,..., X_n$ is called extensive iff $$ A(\lambda X_1,\lambda X_2,...,\lambda X_n)=\lambda A(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \ \ \tag{1}$$ In other words, a quantity $A $ is intensive if it is described by a first order homogeneous multivariable function. On the other hand, a quantity $B$ is intensive iff $$ B(\lambda X_1,\lambda X_2,..., \lambda X_n)=B(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \tag{2}$$ which makes $B$ a $0$ degree homogeneous function.

While the preceding definitions make sense, I don't get why an extensive variable should be a homogeneous function of first degree. For instance, an ideal gas has a potential energy function $$ U(N,T) := \frac{3}{2}Nk_BT,$$ with $N$ being the number of particles and $T$ the temperature of our system. Obviously, the "test" (1) fails for most choices of $ λ $ (except the trivial cases $0$ and $1$) as it is:

$$ U(λN,λT)=λ^2 * U(N,T)$$ which is a second-order homogeneous function. Should one extend $(1)$ to be:

$$ A(λX_1,λX_2,...,λX_n)=λ^kA(X_1,X_2,...,X_n) \ \forall λ\in \mathbb R \ \ (1)$$ for some $k \in \mathbb N^* $? and thus defining an extensive quantity as a homogeneous function of $k$ degree. Are there other rigorous definitions out there?

Most undergraduate books on Thermodynamics offer intuitive definitions for intensive and extensive thermodynamic variables. Authors assert, for example, that the former is independent of the system's size while the latter is not. Other authors argue splitting the system into several subsystems and investigating if the quantity of interest changes or remains the same in those. While all approaches make physical sense, I haven't seen someone define those two concepts in a rigorous manner (even now that I am in university).

In the question Understanding "natural variables" of the thermodynamic potentials using the example of the ideal gas, one person argued in the answers that an extensive quantity, say $A$ which is dependent from other thermodynamic variables $X_1, X_2,..., X_n$ is called extensive iff $$ A(\lambda X_1,\lambda X_2,...,\lambda X_n)=\lambda A(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \ \ \tag{1}$$ In other words, a quantity $A $ is intensive if it is described by a first-order homogeneous multivariable function. On the other hand, a quantity $B$ is intensive iff $$ B(\lambda X_1,\lambda X_2,..., \lambda X_n)=B(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \tag{2}$$ which makes $B$ a $0$ degree homogeneous function.

While the preceding definitions make sense, I don't get why an extensive variable should be a homogeneous function of first degree. For instance, an ideal gas has a potential energy function $$ U(N,T) := \frac{3}{2}Nk_BT,$$ with $N$ being the number of particles and $T$ the temperature of our system. Obviously, the "test" (1) fails for most choices of $ λ $ (except the trivial cases $0$ and $1$) as it is:

$$ U(λN,λT)= \lambda^2 * U(N,T)$$ which is a second-order homogeneous function. Should one extend $(1)$ to be:

$$ A(λX_1,λX_2,...,λX_n)=λ^kA(X_1,X_2,...,X_n) \ \forall λ\in \mathbb R \ \ (1)$$ for some $k \in \mathbb N^* $? and thus defining an extensive quantity as a homogeneous function of $k$ degree. Are there other rigorous definitions out there?

Most undergraduate books on Thermodynamics offer intuitive definitions for intensive and extensive thermodynamic variables. Authors assert, for example, that the former is independent of the system's size while the latter is not. Others argue splitting the system into several subsystems and investigating if the quantity of interest changes or remains the same in those. While all approaches make physical sense, I haven't seen someone defining those two concepts in a rigorous manner (even now that I am in university).

In the question Understanding "natural variables" of the thermodynamic potentials using the example of the ideal gas, one person argued in the answers that an extensive quantity, say $A$ which is dependent from other thermodynamic variables $X_1, X_2,..., X_n$ is called extensive iff $$ A(\lambda X_1,\lambda X_2,...,\lambda X_n)=\lambda A(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \ \ \tag{1}$$ In other words, a quantity $A $ is intensive if it is described by a first order homogeneous multivariable function. On the other hand, a quantity $B$ is intensive iff $$ B(\lambda X_1,\lambda X_2,..., \lambda X_n)=B(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \tag{2}$$ which makes $B$ a $0$ degree homogeneous function.

While the preceding definitions make sense, I don't get why an extensive variable should be a homogeneous function of first degree. For instance, an ideal gas has a potential energy function $$ U(N,T) := \frac{3}{2}Nk_BT,$$ with $N$ being the number of particles and $T$ the temperature of our system. Obviously, the 'test' (1) fails for most choices of $ λ $ (except the trivial cases $0$ and $1$) as it is:

$$ U(λN,λT)=λ^2 * U(N,T)$$ which is a second-order homogeneous function. Should one extend $(1)$ to be:

$$ A(λX_1,λX_2,...,λX_n)=λ^kA(X_1,X_2,...,X_n) \ \forall λ\in \mathbb R \ \ (1)$$ for some $k \in \mathbb N^* $? and thus defining an extensive quantity as a homogeneous function of $k$ degree. Are there other rigorous definitions out there?

Most undergraduate books on Thermodynamics offer intuitive definitions for intensive and extensive thermodynamic variables. Authors assert, for example, that the former is independent of the system's size while the latter is not. Other authors argue splitting the system into several subsystems and investigating if the quantity of interest changes or remains the same in those. While all approaches make physical sense, I haven't seen someone defining those two concepts in a rigorous manner (even now that I am in university).

In the question Understanding "natural variables" of the thermodynamic potentials using the example of the ideal gas, one person argued in the answers that an extensive quantity, say $A$ which is dependent from other thermodynamic variables $X_1, X_2,..., X_n$ is called extensive iff $$ A(\lambda X_1,\lambda X_2,...,\lambda X_n)=\lambda A(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \ \ \tag{1}$$ In other words, a quantity $A $ is intensive if it is described by a first order homogeneous multivariable function. On the other hand, a quantity $B$ is intensive iff $$ B(\lambda X_1,\lambda X_2,..., \lambda X_n)=B(X_1,X_2,...,X_n) \ \forall \lambda \in \mathbb R \tag{2}$$ which makes $B$ a $0$ degree homogeneous function.

While the preceding definitions make sense, I don't get why an extensive variable should be a homogeneous function of first degree. For instance, an ideal gas has a potential energy function $$ U(N,T) := \frac{3}{2}Nk_BT,$$ with $N$ being the number of particles and $T$ the temperature of our system. Obviously, the "test" (1) fails for most choices of $ λ $ (except the trivial cases $0$ and $1$) as it is:

$$ U(λN,λT)=λ^2 * U(N,T)$$ which is a second-order homogeneous function. Should one extend $(1)$ to be:

$$ A(λX_1,λX_2,...,λX_n)=λ^kA(X_1,X_2,...,X_n) \ \forall λ\in \mathbb R \ \ (1)$$ for some $k \in \mathbb N^* $? and thus defining an extensive quantity as a homogeneous function of $k$ degree. Are there other rigorous definitions out there?