CalculusLesson 2 of 3 in the course First-order ordinary differential equations

Exact Differentials: Potentials, Path Independence, Tests, and Derivative Rules

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

Save course
Lesson progress
0 / 5

Your progress is kept in this browser only.Sign up to save it

Summary

An exact differential is a differential form that comes from differentiating a scalar function. Recognizing this structure lets us recover a potential function and understand how changes in a system relate to its variables.

A differential form in three variables can be written as

A(x,y,z) dx+B(x,y,z) dy+C(x,y,z) dz.A(x,y,z)\,dx+B(x,y,z)\,dy+C(x,y,z)\,dz.

dQ=∂Q∂x dx+∂Q∂y dy+∂Q∂z dz=A dx+B dy+C dz.dQ=\frac{\partial Q}{\partial x}\,dx+\frac{\partial Q}{\partial y}\,dy+\frac{\partial Q}{\partial z}\,dz=A\,dx+B\,dy+C\,dz.

All flashcards

Exact differential
A differential form equal to the total differential of a differentiable scalar potential on its domain.
Path independence
The property that a line integral depends only on its initial and final points.
State function
A thermodynamic quantity determined by the system's current equilibrium state, not its path.
Two-dimensional exactness test
For a sufficiently smooth exact form, the cross partials satisfy ∂A/∂y=∂B/∂x\partial A/\partial y=\partial B/\partial x.
Reciprocity relation
Inverse partial derivatives with the same held-fixed variable multiply to one.

Quiz questions

  1. If A dx+B dyA\,dx+B\,dy is exact and its coefficients have continuous first partial derivatives, which condition must hold?Answer: ∂A/∂y=∂B/∂x\partial A/\partial y=\partial B/\partial x
  2. What is the line integral of an exact differential around a closed path?Answer: Zero, because the initial and final potential values are equal.
  3. Under suitable smoothness conditions, what follows when a vector field has zero curl on a simply connected open domain?Answer: The field is conservative and has a potential.
  4. If xx and zz are locally inverse functions of each other while yy is fixed, what is (∂z/∂x)y(∂x/∂z)y\left(\partial z/\partial x\right)_y\left(\partial x/\partial z\right)_y?Answer: 11
  5. For three variables constrained by one differentiable relation, what is the cyclic product (∂x/∂y)z(∂y/∂z)x(∂z/∂x)y\left(\partial x/\partial y\right)_z\left(\partial y/\partial z\right)_x\left(\partial z/\partial x\right)_y?Answer: −1-1

Based on the Wikipedia article Exact differential, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

More in Calculus

All Calculus lessons
CalculusRatio Tests and Their Refinements for Infinite Series5 flashcards · 5 quiz questions · 4 gamesPlay free →CalculusMultiple Integrals: Iterated Methods, Coordinate Changes, and Applications5 flashcards · 5 quiz questions · 4 gamesPlay free →CalculusDivergence: Flux, Coordinate Formulas, and Differential Identities5 flashcards · 5 quiz questions · 4 gamesPlay free →CalculusIntegral Substitution, Jacobians, and Probability Densities5 flashcards · 5 quiz questions · 4 gamesPlay free →CalculusIntegration by Parts: Formulas, Strategies, and Applications5 flashcards · 5 quiz questions · 4 gamesPlay free →CalculusProduct Rule: Proof, Higher Derivatives, and Extensions5 flashcards · 5 quiz questions · 4 gamesPlay free →