Opening page to Rudolf Clausius' 1875 Mathematical Introduction chapter. |

“Every force tends to give motion to the body on which it acts; but it may be prevented from doing so by other opposing forces, so that equilibrium results, and the body remains at rest. In this case the force performs no work. But as soon as the body moves under the influence of the force, work is performed.”

The mathematical introduction originated in the 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable” published in Dingler’s

In this section, Clausius introduces the basics of French physicist Gustave Coriolis’ 1829 principle of the transmission of work (although he dosen't site Coriolis), in that whenever a body moves under the influence of a force, work is performed, and that this movement can be quantified as a product of the line of motion of the particle and the component of the force in the direction of motion.

In this section, Clausius introduces the basic differential work equation:

wheredW = S ds(1)

dW = X dx + Y dy(3)

Clausius here introduces the reader to the elusive importance of the “complete differential”, whereby to be a complete differential, the functions

(4)

If the the condition of equation (4) is satisfied then, according to Clausius, the expression on the right of equation (3) becomes immediately integrable. This section is very important in that it sets the definition of the state variable.

(add)

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Here, Clausius famous introduces readers to the definition of the energy (or internal energy, in a modern terms), symbol U, of the system:

U = T + J

where

References

1. Clausius, Rudolf. (1875).

2. (a) Clausius, Rudolf. (1858). “On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s

(b) Clausius, Rudolf. (1865).