Conservative Vector Fields

A conservative vector field is the gradient of some function f, called a potential function: F = ∇f. For these fields, a line integral depends only on the endpoints, because ∫_C ∇f · dr = f(end) − f(start). In the plane, F = ⟨P, Q⟩ is conservative on an open region without holes exactly when ∂P/∂y = ∂Q/∂x.

Before this lesson: Line Integrals, Vector Fields and Partial Derivatives

Why some fields are conservative

In line integrals, the work along a path usually depends on the path you take. Some force fields are special: gravity near the ground does the same work on a falling ball whether it drops straight down or rolls down a twisting ramp. Only the change in height matters.

Fields like this are called conservative because mechanical energy is conserved in them. The function f plays the role of energy, and the work is just the change in f. If you go around a closed loop and come back to the start, the work is 0.

Fundamental theorem for line integrals

\int_C \nabla f \cdot d\mathbf{r} = f\big(\mathbf{r}(b)\big) - f\big(\mathbf{r}(a)\big)

This works for any smooth path C from r(a) to r(b). It is the line-integral version of the fundamental theorem of calculus.

Test for a conservative field

\mathbf{F} = \langle P, Q \rangle \text{ is conservative} \iff \dfrac{\partial P}{\partial y} = \dfrac{\partial Q}{\partial x}

The test needs P and Q to have continuous partial derivatives on an open region with no holes (a simply connected region), such as the whole plane. In space, F = ⟨P, Q, R⟩ is conservative when P_y = Q_x, P_z = R_x and Q_z = R_y.

1. To find a potential function f, integrate P with respect to x: f = ∫ P dx + g(y).
2. Differentiate that f with respect to y and set it equal to Q.
3. Solve for g′(y), then integrate to find g(y).
4. Write f. Any constant can be added; it cancels in f(end) − f(start).

Conservative vector fields examples

Example 1: Test a field and find its potential function

Show that F(x, y) = ⟨2xy + 3, x² − 4y⟩ is conservative and find a potential function.

P_y = \dfrac{\partial}{\partial y}(2xy + 3) = 2x

Q_x = \dfrac{\partial}{\partial x}(x^2 - 4y) = 2x

P_y = Q_x

f = \int (2xy + 3)\,dx = x^2y + 3x + g(y)

f_y = x^2 + g'(y)

x^2 + g'(y) = x^2 - 4y

g'(y) = -4y

g(y) = -2y^2

f(x, y) = x^2y + 3x - 2y^2

Answer: f(x, y) = x²y + 3x − 2y² (plus any constant). Check: f_x = 2xy + 3 and f_y = x² − 4y.

Example 2: Use the fundamental theorem

For F from Example 1, evaluate \int_C \mathbf{F} \cdot d\mathbf{r}, where C is any path from (1, 0) to (2, 1).

\int_C \mathbf{F} \cdot d\mathbf{r} = f(2, 1) - f(1, 0)

f(2, 1) = 4 + 6 - 2 = 8

f(1, 0) = 0 + 3 - 0 = 3

8 - 3 = 5

Answer: 5, for every path from (1, 0) to (2, 1). No parametrization is needed.

Example 3: A field that is not conservative

Show that F(x, y) = ⟨y, 2x⟩ is not conservative. Then compare ∫_C F · dr from (0, 0) to (1, 1) along y = x and along y = x².

P_y = 1, \quad Q_x = 2

P_y \ne Q_x

\mathbf{r} = \langle t, t \rangle: \; \int_0^1 (t + 2t)\,dt = \dfrac{3}{2}

\mathbf{r} = \langle t, t^2 \rangle: \; \int_0^1 (t^2 + 2t \cdot 2t)\,dt

= \int_0^1 5t^2\,dt = \dfrac{5}{3}

Answer: the two paths give 3/2 and 5/3. The value depends on the path, as expected for a field that is not conservative.

Example 4: A conservative field in space

Show that F(x, y, z) = ⟨2x, z², 2yz + 1⟩ is conservative, find a potential function, and evaluate ∫_C F · dr from (0, 0, 0) to (1, 2, 3).

P_y = 0 = Q_x, \quad P_z = 0 = R_x

Q_z = 2z = R_y

f = \int 2x\,dx = x^2 + g(y, z)

g_y = z^2 \Rightarrow g = yz^2 + h(z)

2yz + h'(z) = 2yz + 1 \Rightarrow h(z) = z

f = x^2 + yz^2 + z

f(1, 2, 3) - f(0, 0, 0) = (1 + 18 + 3) - 0 = 22

Answer: f = x² + yz² + z, and the line integral is 22 along any path.

Example 5: Work in a force field

A robot arm moves a part through the force field F(x, y) = ⟨2x + y, x + 4y⟩ newtons, with x and y in meters. It moves from A(1, 1) to B(3, 2) along a winding path. How much work does the field do? How much work does it do on a trip from A to B and back to A?

P_y = 1 = Q_x

f = \int (2x + y)\,dx = x^2 + xy + g(y)

x + g'(y) = x + 4y \Rightarrow g(y) = 2y^2

f = x^2 + xy + 2y^2

f(3, 2) = 9 + 6 + 8 = 23

f(1, 1) = 1 + 1 + 2 = 4

W = 23 - 4 = 19

Answer: the field does 19 J of work, whatever the shape of the path. On a round trip back to A the total work is f(A) − f(A) = 0 J.

Common mistakes

Related methods

Closed curves

For a conservative field, ∮_C F · dr = 0 around every closed curve C. If you find one closed curve with a nonzero result, the field is not conservative. Green’s theorem, the next lesson, connects this to Q_x − P_y.

Finding f by inspection

With practice you can often spot f. For example, ⟨y, x⟩ is ∇(xy) and ⟨x, y⟩ is ∇((x² + y²)/2). Always check by taking the gradient of your answer.

Energy in physics

Physicists write F = −∇U, where U is the potential energy. Then the work done by F from A to B is U(A) − U(B), the drop in potential energy.

Practice problems

1. Is F = ⟨3x², 2y⟩ conservative? If so, find a potential function.
2. Is F = ⟨x + y, x − y⟩ conservative? If so, find a potential function.
3. Is F = ⟨xy, x + y⟩ conservative?
4. Find a potential function for F = ⟨eʸ, xeʸ + 2y⟩.
5. Evaluate ∫_C ⟨2x, 2y⟩ · dr along any path from (1, 2) to (3, −1).
6. Evaluate ∫_C ⟨y cos x, sin x⟩ · dr along any path from (0, 1) to (π/2, 3).
7. Evaluate ∮_C ⟨2xy, x²⟩ · dr once around the unit circle.
8. Evaluate ∫_C ⟨yz, xz, xy⟩ · dr along any path from (1, 1, 1) to (2, 3, −1).
9. Show that F = ⟨x, y, z⟩ is conservative and evaluate ∫_C F · dr from (1, 0, 0) to (0, 3, 4).
10. A 5 kg box is carried up a spiral staircase, rising 12 m. Gravity is F = ⟨0, 0, −49⟩ newtons. Find the work done by gravity.

P_y = 0 = Q_x

f = \int 3x^2\,dx = x^3 + g(y), \quad g'(y) = 2y

f = x^3 + y^2

P_y = 1 = Q_x

f = \dfrac{x^2}{2} + xy + g(y), \quad x + g'(y) = x - y

g(y) = -\dfrac{y^2}{2}

P_y = x, \quad Q_x = 1

x \ne 1 \text{ in general}

f = \int e^y\,dx = xe^y + g(y)

xe^y + g'(y) = xe^y + 2y \Rightarrow g(y) = y^2

f = x^2 + y^2

f(3, -1) - f(1, 2) = 10 - 5 = 5

f = y\sin x

f\Big(\dfrac{\pi}{2}, 3\Big) - f(0, 1) = 3(1) - 1(0) = 3

P_y = 2x = Q_x

\oint_C \mathbf{F} \cdot d\mathbf{r} = 0

f = xyz

f(2, 3, -1) - f(1, 1, 1) = -6 - 1 = -7

f = \dfrac{x^2 + y^2 + z^2}{2}

f(0, 3, 4) - f(1, 0, 0) = \dfrac{25}{2} - \dfrac{1}{2} = 12

f = -49z

W = -49(12) - (-49)(0) = -588

Conservative vector fields FAQ

How do you know if a vector field is conservative?

For F = ⟨P, Q⟩ on a region without holes, check whether ∂P/∂y = ∂Q/∂x. In 3D, check the three pairs P_y = Q_x, P_z = R_x and Q_z = R_y, which is the same as curl F = 0.

What is a potential function?

It is a function f whose gradient is the field: ∇f = F. It is unique up to an added constant.

What does path independence mean?

It means ∫_C F · dr has the same value for every path with the same starting and ending points. A field is path independent exactly when it is conservative.

Why is the line integral of a conservative field around a closed curve zero?

By the fundamental theorem for line integrals, the integral equals f(end) − f(start). On a closed curve the end and start are the same point, so the difference is 0.

Summary

• F is conservative when F = ∇f for a potential function f.
• Fundamental theorem: ∫_C ∇f · dr = f(end) − f(start), for any path.
• Test in 2D: P_y = Q_x on a region without holes. In 3D: P_y = Q_x, P_z = R_x, Q_z = R_y.
• Find f by integrating P in x, then matching f_y with Q.
• Around any closed curve, a conservative field does zero work.

Previous lesson: Line Integrals

Next lesson: Green’s Theorem

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