修改体积导数计算逻辑，补充完善index.md

tests/cryo_tank/test_properties.py

@@ -6,6 +6,7 @@ sys.path.insert(0, "src")
 
 from cryo_tank.properties import (
     ln2_rho, ln2_h, ln2_u, ln2_T_from_u,
+    ln2_drho_dT_const_p, ln2_du_dT_const_p,
     he_u, he_h, he_cp, he_cv,
 )
 from cryo_tank.config import P_WORKING
@@ -32,6 +33,21 @@ class TestLN2Properties:
         T_recovered = ln2_T_from_u(u, P_WORKING)
         assert abs(T_recovered - T_orig) < 0.01
 
+    def test_ln2_density_derivative_matches_finite_difference(self):
+        T = 78.0
+        step = 1e-3
+        actual = ln2_drho_dT_const_p(T, P_WORKING)
+        expected = (
+            ln2_rho(T + step, P_WORKING)
+            - ln2_rho(T - step, P_WORKING)
+        ) / (2.0 * step)
+        assert actual < 0.0
+        assert abs(actual - expected) / abs(expected) < 1e-5
+
+    def test_ln2_internal_energy_derivative_is_positive(self):
+        du_dT = ln2_du_dT_const_p(78.0, P_WORKING)
+        assert du_dT > 0.0
+
 
 class TestHeliumProperties:
     """Helium properties."""