Exact differential

Definition

An exact differential is a differential form that can be expressed as the gradient of some scalar function. It indicates a path-independent process in thermodynamics and mechanics.

5 Must Know Facts For Your Next Test

1. An exact differential implies the existence of a potential function such that $dU = \frac{\partial U}{\partial x} dx + \frac{\partial U}{\partial y} dy$.
2. For a force field to be conservative, its work done must equal the change in a potential function, represented by an exact differential.
3. A necessary condition for a differential $M(x,y)dx + N(x,y)dy$ to be exact is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
4. If the curl of a vector field is zero, then the field is conservative, and its line integral around any closed loop is zero.
5. Exact differentials are often used in identifying conservative forces where the total mechanical energy remains constant.

Review Questions

• What condition must be met for a differential $M(x,y)dx + N(x,y)dy$ to be considered exact?
• How does an exact differential relate to conservative forces and potential energy?
• Explain why the curl of a vector field being zero signifies that it is an exact differential.

Related terms

Conservative Force: A force with the property that the work done in moving an object between two points is independent of the path taken.

Potential Energy: The energy stored by an object due to its position in a force field, commonly gravitational or elastic.

Gradient: \text{A vector operator that shows how much and in what direction something changes most rapidly.}