Editing Exact differential (section)

=== Integral path independence ===
The exact differential for a differentiable scalar function <math>Q</math> defined in an open domain <math>D \subset \mathbb{R}^n</math> is equal to <math>dQ = \nabla Q \cdot d \mathbf{r}</math>, where <math>\nabla Q</math> is the [[gradient]] of <math>Q</math>, <math>\cdot</math> represents the [[Dot product|scalar product]], and <math>d \mathbf{r}</math> is the general differential displacement vector, if an orthogonal coordinate system is used. If <math>    Q</math> is of differentiability class <math>C^1</math> ([[Smoothness#Multivariate differentiability classes|continuously differentiable]]), then <math>\nabla Q</math> is a [[conservative vector field]] for the corresponding potential <math>Q</math> by the definition. For three dimensional spaces, expressions such as <math>d \mathbf{r} = (dx, dy, dz)</math> and <math>\nabla Q = \left(\frac{\partial Q}{\partial x}, \frac{\partial Q}{\partial y},\frac{\partial Q}{\partial z}\right)</math> can be made.

The [[gradient theorem]] states

:<math>\int _{i}^{f} dQ = \int _{i}^{f}\nabla Q (\mathbf {r} )\cdot d \mathbf {r} = Q \left(f \right) - Q \left(i \right)</math>

that does not depend on which integral path between the given path endpoints <math>i</math> and <math>f</math> is chosen. So it is concluded that ''the integral of an exact differential is independent of the choice of an integral path between given path endpoints [[Conservative vector field#Path independence|(path independence)]].''

For three dimensional spaces, if <math>\nabla Q</math> defined on an open domain <math>D \subset \mathbb{R}^3</math> is of [[Smoothness#Multivariate differentiability classes|differentiability class]] <math>C^1</math> (equivalently <math>Q</math> is of <math>C^2</math>), then this integral path independence can also be proved by using the [[Vector calculus identity#Curl of gradient is zero|vector calculus identity]] <math>\nabla \times ( \nabla Q )  = \mathbf{0}</math> and [[Stokes' theorem]].

:<math>\oint _{\partial \Sigma }\nabla Q \cdot d \mathbf {r} = \iint _{\Sigma }(\nabla \times \nabla Q)\cdot d \mathbf {a} = 0</math>
for a simply closed loop <math>\partial \Sigma</math> with the smooth oriented surface <math>\Sigma</math> in it. If the open domain <math>D</math> is [[Simply connected space|simply connected open space]] (roughly speaking, a single piece open space without a hole within it), then any irrotational vector field (defined as a <math>C^1</math> vector field <math>\mathbf{v}</math> which curl is zero, i.e., <math>\nabla \times \mathbf{v}  = \mathbf{0}</math>) has the path independence by the Stokes' theorem, so the following statement is made; ''In a simply connected open region, any'' <math>C^1</math> ''vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vice versa.'' The equality of the path independence and conservative vector fields is shown [[Conservative vector field#Path independence and conservative vector field|here]].

==== Thermodynamic state function ====
In [[thermodynamics]], when <math>dQ</math> is exact, the function <math>Q</math> is a [[state function]] of the system: a [[Function (mathematics)|mathematical function]] which depends solely on the current [[Thermodynamic equilibrium|equilibrium state]], not on the path taken to reach that state. [[Internal energy]] <math>U</math>, [[Entropy]] <math>S</math>, [[Enthalpy]] <math>H</math>, [[Helmholtz free energy]] <math>A</math>, and [[Gibbs free energy]] <math>G</math> are [[state function]]s. Generally, neither [[Work (thermodynamics)|work]] <math>W</math> nor [[heat]] <math>Q</math> is a state function. (Note: <math>Q</math> is commonly used to represent heat in physics. It should not be confused with the use earlier in this article as the parameter of an exact differential.)