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Exact differential

In mathematics and physics, an exact differential is a differential form df = P(\mathbf{x}) \cdot d\mathbf{x} in multiple variables that arises as the total differential of a scalar potential function f, such that P_i = \frac{\partial f}{\partial x_i} for each component, ensuring the line integral \int df between two points is path-independent and equals f(B) - f(A).^[1] For a two-variable case, df = P(x,y) \, dx + Q(x,y) \, dy is exact if there exists f(x,y) satisfying these partial derivative relations.^[2] A key test for exactness, assuming the functions are continuously differentiable, is the equality of mixed partial derivatives: \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}.^[1] This condition stems from Clairaut's theorem on the equality of mixed partials and guarantees the existence of the potential function locally.^[2] Exact differentials play a central role in thermodynamics, where they distinguish state functions—like internal energy U or entropy S, whose differentials dU and dS are exact—from path functions like heat \delta Q and work \delta W, which are inexact and depend on the process path.^[2] In this context, exactness implies the function's value depends only on the system's state, not its history, enabling the formulation of fundamental relations such as dU = \delta Q - \delta W.^[2] In the study of differential equations, an equation M(x,y) \, dx + N(x,y) \, dy = 0 is termed exact if its left side is the exact differential of some function \Psi(x,y), allowing direct integration to yield the implicit solution \Psi(x,y) = C.^[3] This approach simplifies solving first-order equations without needing integrating factors, provided the exactness condition holds.^[3]

Definition and Properties

Formal Definition

In multivariable calculus, an exact differential is a differential form that arises as the total differential of some scalar potential function. Consider a differential form in two variables, \omega = P(x,y)\, dx + Q(x,y)\, dy. This form is exact if there exists a scalar function f(x,y) such that df = \omega, meaning \frac{\partial f}{\partial x} = P(x,y) and \frac{\partial f}{\partial y} = Q(x,y).^[1]^[4] A necessary and sufficient condition for exactness in two variables, assuming sufficient smoothness of P and Q, is that the mixed partial derivatives satisfy \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}. This equality follows from Clairaut's theorem (also known as Schwarz's theorem), which states that if the second partial derivatives of f are continuous, then \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}.^[1]^[4]^[5] This concept generalizes to n variables, where a 1-form \omega = \sum_{i=1}^n P_i(\mathbf{x})\, dx_i is exact if it is the exterior derivative of a 0-form (scalar function), or equivalently, if the associated vector field \mathbf{P} = (P_1, \dots, P_n) is conservative; in particular, in three dimensions, its curl vanishes: \nabla \times \mathbf{P} = \mathbf{0}.^[6]^[7] The formalization of exact differentials within the framework of vector calculus occurred in the late 19th century, primarily through the independent contributions of Josiah Willard Gibbs and Oliver Heaviside, who developed the modern notation and theorems for vector fields and their differentials.^[8]

Path Independence

A defining characteristic of an exact differential \omega is that the line integral \int_C \omega between two fixed endpoints a and b in the domain is independent of the path C connecting them.^[9] This property holds because an exact differential satisfies \omega = df for some scalar potential function f, ensuring the integral captures only the net change in f.^[10] The fundamental theorem for line integrals formalizes this path independence. For a smooth curve C parameterized by \vec{r}(t) with a \leq t \leq b, and \omega = \nabla f \cdot d\vec{r} where \nabla f is continuous, the theorem states:

\int_C \nabla f \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)).

This equality depends solely on the endpoints, not the specific path.^[9] To derive this, parameterize the curve and apply the chain rule:

\nabla f \cdot \frac{d\vec{r}}{dt} = \frac{d}{dt} [f(\vec{r}(t))].

Integrating both sides from t = a to t = b and invoking the fundamental theorem of calculus yields:

\int_a^b \frac{d}{dt} [f(\vec{r}(t))] \, dt = f(\vec{r}(b)) - f(\vec{r}(a)).

Thus, the line integral simplifies to the difference in the potential function values, confirming path independence.^[10] In contrast, inexact differentials lead to path-dependent integrals, where the value varies with the chosen route. For instance, the work done by non-conservative forces, such as friction, depends on the trajectory taken, as energy dissipation accumulates differently along varied paths.^[11]^[12] A simple example illustrates this for an exact differential. Consider \omega = 2xy \, dx + x^2 \, dy, which is exact with potential f(x,y) = x^2 y. The line integral from (0,0) to (1,1) equals f(1,1) - f(0,0) = 1 - 0 = 1, regardless of path. Direct computation along the straight line y = x (where dy = dx, $0 \leq x \leq 1) gives:

\int_0^1 (2x \cdot x + x^2 \cdot 1) \, dx = \int_0^1 (3x^2) \, dx = 1.

Along the piecewise path (first along the x-axis to (1,0), then up the y-axis to (1,1)), the integral is $0 + \int_0^1 x^2 \, dy = 1 (with x=1), confirming the same result.^[9]

Thermodynamic Applications

State Functions

In thermodynamics, state functions are properties of a system that depend solely on its current state, defined by variables such as temperature, pressure, and volume, rather than the history or path taken to reach that state. Examples include internal energy U, enthalpy H, entropy S, Helmholtz free energy F, and Gibbs free energy G. In contrast, process functions like heat Q and work W are path-dependent, meaning their values vary with the specific process connecting initial and final states.^[13] The differential of a state function is exact, ensuring that changes in the function are independent of the path. For instance, the first law of thermodynamics states that dU = \delta Q - \delta W, where \delta Q and \delta W are inexact differentials for infinitesimal heat and work transfers (with \delta W denoting work done by the system), but their difference dU is exact. This holds for any process, reversible or irreversible, implying that the finite change \Delta U = U_{\text{final}} - U_{\text{initial}} depends only on the initial and final states, not the intermediate path.^[14] A key criterion for identifying state functions is that the line integral of their differential around any closed cycle vanishes: \oint d\phi = 0, where \phi is the state function. This property confirms path independence and distinguishes state functions from process functions, whose cyclic integrals are generally nonzero.^[15]^[16] In modern contexts, such as non-equilibrium thermodynamics, the exactness of differentials for state functions may not hold in the traditional sense, as systems deviate from equilibrium states during irreversible processes like rapid expansions or transport phenomena. Here, internal energy and other state variables can exhibit path-dependent behaviors due to entropy production and spatial gradients, requiring extensions like local equilibrium approximations or additional flux variables to describe dynamics accurately.^[17]

Examples in Thermodynamics

In thermodynamics, the internal energy U serves as a fundamental example of an exact differential, expressed as dU = T \, dS - P \, dV, where T is temperature, S is entropy, P is pressure, and V is volume. This form is exact because U is a state function depending solely on the state variables S and V, ensuring that changes in U are path-independent and determined only by the initial and final states of the system.^[18]^[19] Similarly, the enthalpy H, defined as H = U + PV, has the exact differential dH = T \, dS + V \, dP. As a state function with natural variables S and P, H exhibits path independence, meaning the change \Delta H for any process depends only on the endpoints, making it particularly useful for constant-pressure processes where \Delta H equals the heat transferred.^[18]^[19] The path independence of exact differentials like those for U and H highlights a key distinction in thermodynamic processes: reversible versus irreversible. In reversible processes, the equality dU = T \, dS - P \, dV holds exactly, allowing full recovery of work and heat along the path, whereas irreversible processes involve inequalities (e.g., dq < T \, dS), but the net change in state functions such as \Delta U or \Delta H remains path-independent, relying only on initial and final states.^[20] The Gibbs free energy G = H - TS provides another illustration, with its exact differential dG = -S \, dT + V \, dP, exact due to its dependence on state variables T and P. This form is crucial in phase equilibria, where at constant T and P, the minimum G determines stable phases, and equilibrium between phases occurs when their chemical potentials are equal, ensuring dG = 0.^[19]^[21]

Mathematical Formulations

One-Dimensional Case

In the one-dimensional case, the differential of a function f(x) is given by df = f'(x)\, dx, where f'(x) is the derivative of f with respect to x. This form is exact by definition, as it directly represents the infinitesimal change in f corresponding to an infinitesimal change dx in the independent variable x.^[22] The integral of this exact differential from a point a to b yields \int_a^b df = f(b) - f(a), which follows from the Fundamental Theorem of Calculus and holds regardless of the specific path taken, since in one dimension there is only a single possible path along the line.^[22] This path independence is a trivial consequence of the linear nature of one-dimensional space. A representative example from mechanics illustrates this concept: the position s(t) of a particle moving along a straight line serves as the potential function whose exact differential is the displacement ds = v(t)\, dt, where v(t) is the velocity. Integrating this differential gives the change in position \Delta s = \int_{t_1}^{t_2} v(t)\, dt = s(t_2) - s(t_1), directly linking the antiderivative of velocity to displacement.^[23] This one-dimensional formulation is fundamentally tied to the concept of antiderivatives in calculus, where the exact differential df = f'(x)\, dx implies that f(x) is the antiderivative of f'(x), up to a constant, ensuring that integration recovers the original function precisely.^[22]

Multidimensional Case

In the multidimensional case, the concept of an exact differential extends beyond one variable to differential forms on \mathbb{R}^n, where a 1-form \omega = P \, dx_1 + Q \, dx_2 + \cdots + R \, dx_n is exact if there exists a scalar potential function f such that \omega = df, meaning the coefficients are the partial derivatives of f.^[24] This corresponds to the vector field (P, Q, \dots, R) being conservative, with \nabla f = (P, Q, \dots, R).^[25] In two dimensions, consider the 1-form \omega = P(x,y) \, dx + Q(x,y) \, dy. The form is exact if and only if it is closed, satisfying the condition \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}, which is equivalent to the curl of the associated vector field (P, Q) being zero: \nabla \times (P, Q) = 0.^[24] This test ensures path independence of the line integral \int_C \omega, a hallmark of conservative fields where the integral depends only on the endpoints.^[25] For three dimensions, the 1-form \omega = P(x,y,z) \, dx + Q(x,y,z) \, dy + R(x,y,z) \, dz is exact if the vector field (P, Q, R) has zero curl: \nabla \times (P, Q, R) = 0.^[26] The components of the curl are \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) = (0, 0, 0).^[24] Again, this implies the line integral \int_C \omega is path-independent, equaling f(B) - f(A) for endpoints A and B.^[25] To find the potential function f for an exact form, integrate one coefficient while treating others as constants, then determine remaining arbitrary functions using the other coefficients. In two dimensions, integrate P with respect to x to obtain f(x,y) = \int P \, dx + h(y), differentiate with respect to y, and set equal to Q to solve for h'(y).^[25] In three dimensions, similarly integrate P with respect to x to get f(x,y,z) = \int P \, dx + g(y,z), then use Q to find \frac{\partial g}{\partial y} and R to find \frac{\partial g}{\partial z}.^[25] Verify by checking \nabla f = (P, Q, R). A classic example is the gravitational field near Earth's surface, approximated as \mathbf{F} = -g \hat{z}, which is conservative since \nabla \times \mathbf{F} = 0, with potential f = g z + C and path-independent work done by the field.^[25] In simply connected domains, such as \mathbb{R}^2 or \mathbb{R}^3 minus isolated points, the Poincaré lemma guarantees that every closed 1-form is exact, ensuring the existence of a potential function under these topological conditions.^[27]

Differential Relations

Reciprocity Relation

In the context of exact differentials, the reciprocity relation arises as a direct consequence of the exactness condition for a differential form. Consider a function Z(x, y) whose total differential is dZ = M(x, y)\, dx + N(x, y)\, dy, where the differential is exact. This exactness requires that the mixed second partial derivatives of Z are equal, specifically \frac{\partial^2 Z}{\partial x \partial y} = \frac{\partial^2 Z}{\partial y \partial x}. Since \frac{\partial M}{\partial y} = \frac{\partial^2 Z}{\partial y \partial x} and \frac{\partial N}{\partial x} = \frac{\partial^2 Z}{\partial x \partial y}, the reciprocity relation follows: \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.^[28]^[29] This relation can be proven using the equality of mixed partial derivatives, a fundamental theorem in multivariable calculus stating that if the second partial derivatives are continuous, then \frac{\partial^2 Z}{\partial x \partial y} = \frac{\partial^2 Z}{\partial y \partial x}. For the exact differential dZ, differentiating M with respect to y yields the left-hand mixed partial, while differentiating N with respect to x yields the right-hand mixed partial; their equality enforces the reciprocity condition. This proof underscores the path independence of exact differentials, as the condition ensures Z is a state function.^[28] In thermodynamics, the reciprocity relation applies to state functions like internal energy U(S, V), whose differential is dU = T\, dS - P\, dV, an exact differential where M = T and N = -P. Applying the relation gives \left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V. This thermodynamic reciprocity links measurable quantities such as temperature and pressure to entropy changes.^[29] The reciprocity relation serves as the foundation for deriving Maxwell's relations, which are additional equalities among thermodynamic partial derivatives obtained by applying the reciprocity condition to various thermodynamic potentials. For instance, from the differential of the Gibbs free energy dG = -S\, dT + V\, dP, reciprocity yields \left( \frac{\partial S}{\partial P} \right)_T = -\left( \frac{\partial V}{\partial T} \right)_P, facilitating experimental determination of thermodynamic properties.^[29]

Cyclic Relation

In multivariable calculus, the cyclic relation, also known as the triple product rule or Euler's chain rule, emerges as a consequence of the exactness of a total differential for interdependent variables x, y, and z, where z = z(x, y). The relation states that

\left( \frac{\partial x}{\partial y} \right)_z \left( \frac{\partial y}{\partial z} \right)_x \left( \frac{\partial z}{\partial x} \right)_y = -1.

This identity holds because the partial derivatives must satisfy consistency conditions for the differentials to be path-independent.^[30] The derivation follows directly from the total differential dz = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy, which is exact if the mixed partial derivatives are equal, i.e., \frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}. To obtain the cyclic form, express y as a function of x and z, yielding dy = \left( \frac{\partial y}{\partial x} \right)_z dx + \left( \frac{\partial y}{\partial z} \right)_x dz. Substituting the expression for dz and collecting terms leads to coefficients that must vanish for arbitrary dx and dy, resulting in the product equaling -1. This ensures the differential form is integrable and the variables are related through a state function.^[30] In thermodynamics, the cyclic relation applies to state functions derived from exact differentials like the internal energy differential dU = T \, dS - P \, dV, where U = U(S, V), T = \left( \frac{\partial U}{\partial S} \right)_V, and P = -\left( \frac{\partial U}{\partial V} \right)_S. For instance, considering the interdependent variables P, V, and T, the relation takes the form

\left( \frac{\partial P}{\partial T} \right)_V \left( \frac{\partial T}{\partial V} \right)_P \left( \frac{\partial V}{\partial P} \right)_T = -1,

which links the coefficients T and P to measurable properties like the thermal expansion coefficient \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P and isothermal compressibility \beta = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T, yielding \left( \frac{\partial P}{\partial T} \right)_V = \frac{\alpha}{\beta}. This form is verified for ideal gases using the equation of state PV = nRT.^[31]^[32] A common misconception is applying the cyclic relation to non-state variables, such as heat Q or work W, whose differentials dQ and dW are inexact and path-dependent; the relation holds only for state functions where the total differential is exact, ensuring the partial derivatives reflect intrinsic dependencies rather than process-specific paths.^[33] The cyclic relation is distinct from but complementary to the reciprocity relation, which involves pairwise symmetry in second derivatives.^[30]

Derived Equations and Identities

In Two Dimensions

In two dimensions, an exact differential takes the form \omega = M(x,y) \, dx + N(x,y) \, dy, where there exists a scalar potential function f(x,y) such that \omega = df.^[3] The total differential of f is given by

df = \frac{\partial f}{\partial x} \, dx + \frac{\partial f}{\partial y} \, dy,

so M = \partial f / \partial x and N = \partial f / \partial y. Differentiating M yields

dM = \frac{\partial M}{\partial x} \, dx + \frac{\partial M}{\partial y} \, dy,

and since M = \partial f / \partial x, the equality of mixed partial derivatives implies \partial M / \partial y = \partial^2 f / \partial y \partial x = \partial N / \partial x. This leads to the key integrability condition for exactness: \partial M / \partial y = \partial N / \partial x.^[3]^[24] If the differential is inexact, an integrating factor \mu(x,y) may exist such that \mu \omega becomes exact, satisfying \partial (\mu M) / \partial y = \partial (\mu N) / \partial x, though the focus here remains on exact cases.^[34] A prominent application arises in thermodynamics, where the enthalpy differential dH = T \, [dS](/page/DS) + V \, [dP](/page/DP) is exact, with T = (\partial H / \partial S)_P and V = (\partial H / \partial P)_S. The exactness condition \partial T / \partial P = \partial V / \partial S then yields the Maxwell relation

\left( \frac{\partial T}{\partial P} \right)_S = \left( \frac{\partial V}{\partial S} \right)_P.

^[35]^[36] The reciprocity relation in two dimensions, stemming from the equality of mixed partials, ensures that the Jacobian matrix of the transformation defined by the partial derivatives is symmetric. For df = M \, dx + N \, dy, the Jacobian

J = \begin{pmatrix} \frac{\partial M}{\partial x} & \frac{\partial M}{\partial y} \\ \frac{\partial N}{\partial x} & \frac{\partial N}{\partial y} \end{pmatrix}

satisfies J^T = J due to \partial M / \partial y = \partial N / \partial x.^[24]

Generalizations to Higher Dimensions

In the context of multivariable calculus and differential geometry, the concept of an exact differential extends naturally to n dimensions through the framework of differential forms. A smooth 1-form \omega on an open subset of \mathbb{R}^n is said to be exact if there exists a smooth function f: \mathbb{R}^n \to \mathbb{R} such that \omega = df; equivalently, \omega is closed if its exterior derivative vanishes, i.e., d\omega = 0. On contractible domains, such as star-shaped open sets in \mathbb{R}^n, the Poincaré lemma guarantees that every closed form is exact, ensuring the existence of a potential function whose gradient yields the form.^[37] This generalizes the two-dimensional condition where exactness implies path-independent line integrals, now holding for higher-dimensional integrals over simply connected regions.^[38] From such exact differentials arise higher-order relations analogous to Maxwell's relations in thermodynamics, derived from the equality of mixed partial derivatives of the potential function. For a thermodynamic potential \Phi depending on n independent variables x_1, \dots, x_n, the exactness of d\Phi = \sum_{i=1}^n y_i \, dx_i (where the y_i are conjugate variables) implies that the Hessian matrix of second partial derivatives is symmetric, yielding \frac{\partial^2 \Phi}{\partial x_i \partial x_j} = \frac{\partial^2 \Phi}{\partial x_j \partial x_i} for all i, j. This produces \binom{n}{2} independent Maxwell-like relations connecting cross derivatives, such as \left( \frac{\partial y_i}{\partial x_j} \right) = \left( \frac{\partial y_j}{\partial x_i} \right), enforcing the symmetry of the Hessian and reducing the number of independent second derivatives from n^2 to \frac{n(n+1)}{2}. These identities facilitate relating measurable quantities like pressure and entropy across multiple state variables.^[39] A concrete example occurs in three-dimensional thermodynamics with the Helmholtz free energy F(T, V, N), where dF = -S \, dT - P \, dV + \mu \, dN and S, P, \mu are entropy, pressure, and chemical potential, respectively. Exactness leads to cross-partial equalities, including \left( \frac{\partial S}{\partial V} \right)_{T,N} = \left( \frac{\partial P}{\partial T} \right)_{V,N}, \left( \frac{\partial S}{\partial N} \right)_{T,V} = -\left( \frac{\partial \mu}{\partial T} \right)_{V,N}, and \left( \frac{\partial P}{\partial N} \right)_{T,V} = \left( \frac{\partial \mu}{\partial V} \right)_{T,N}, enabling derivations of phase behavior and response functions in multi-component systems.^[35] The condition for closedness in n dimensions corresponds to the vanishing of the n-dimensional analogue of the curl for the associated vector field. For a 1-form \omega = \sum_{i=1}^n F_i \, dx_i, d\omega = 0 implies that the 2-form components satisfy \sum_{i<j} \left( \frac{\partial F_j}{\partial x_i} - \frac{\partial F_i}{\partial x_j} \right) dx_i \wedge dx_j = 0, or in vector terms, the antisymmetric part of the Jacobian vanishes pairwise. However, on non-contractible manifolds, closed forms need not be exact; de Rham cohomology quantifies this obstruction, with cohomology groups H^k(M) measuring the dimension of closed k-forms modulo exact ones, as seen in examples like the punctured plane where the angular form d\theta is closed but not exact.^[37]^[38]

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[37]
[PDF] Differential Forms and Integration - UCLA Mathematics
The concept of a closed form corresponds to that of a conservative force in physics (and an exact form corresponds to the concept of having a potential function) ...
[38]
https://www.math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf
[39]
[PDF] The Maxwell relations
The Maxwell relations. A number of second derivatives of the fundamental relation have clear physical significance and can be measured experimentally. For ...Missing: exact | Show results with:exact