Conservative field in the context of Vector calculus

The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. Without a thermodynamic process, the internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics. The notion has been introduced to describe the systems characterized by temperature variations, temperature being added to the set of state parameters, the position variables known in mechanics (and their conjugated generalized force parameters), in a similar way to potential energy of the conservative fields of force, gravitational and electrostatic. Its author is Rudolf Clausius. Without transfer of matter, internal energy changes equal the algebraic sum of the heat transferred and the work done. In systems without temperature changes, internal energy changes equal the work done by/on the system.

The internal energy cannot be measured absolutely. Thermodynamics concerns changes in the internal energy, not its absolute value. The processes that change the internal energy are transfers, into or out of the system, of substance, or of energy, as heat, or by thermodynamic work. These processes are measured by changes in the system's properties, such as temperature, entropy, volume, electric polarization, and molar constitution. The internal energy depends only on the internal state of the system and not on the particular choice from many possible processes by which energy may pass into or out of the system. It is a state variable, a thermodynamic potential, and an extensive property.

In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the conservative gravitational field. It is analogous to the electric potential with mass playing the role of charge. The reference point, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance. Their similarity is correlated with both associated fields having conservative forces.

Mathematically, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies.