Training Multivariable Calculus Vector Fields & Line Integrals
4 / 5

Vector Fields & Line Integrals

26 min Multivariable Calculus
A vector field F = (P, Q) assigns a vector to each point in the plane, and a line integral ∫_C F · dr measures the accumulated 'work' or 'flux' along a curve C. When F is conservative—meaning F = ∇φ for some potential function φ—the line integral depends only on the endpoints, not on the path. Green's theorem connects a line integral around a closed curve to a double integral over the enclosed region: ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dA. This powerful bridge between line and area integrals generalizes in three dimensions to Stokes' theorem and the divergence theorem, forming the backbone of electromagnetism, fluid dynamics, and differential geometry.

Vector Fields & Line Integrals

Vector Field

A function $\mathbf{F}(x,y) = \langle P(x,y), Q(x,y) \rangle$ that assigns a vector to each point in a region.

Line Integral of a Scalar

$$\int_C f\,ds = \int_a^b f(\mathbf{r}(t)) \|\mathbf{r}'(t)\|\,dt$$

Line Integral of a Vector Field

$$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt$$

This computes the work done by $\mathbf{F}$ along curve $C$.

Conservative Fields

$\mathbf{F}$ is conservative if $\mathbf{F} = \nabla f$ for some potential $f$. Then:

$$\int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A)$$

Test: $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$.

Example 1

Evaluate $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F} = \langle y, x \rangle$ and $C$ is $\mathbf{r}(t) = \langle t, t^2 \rangle$, $0 \leq t \leq 1$.

  1. $\mathbf{r}'(t) = \langle 1, 2t \rangle$.
  2. $\mathbf{F}(\mathbf{r}(t)) = \langle t^2, t \rangle$.
  3. $\int_0^1 (t^2 \cdot 1 + t \cdot 2t)\,dt = \int_0^1 3t^2\,dt = 1$.
Example 2

Is $\mathbf{F} = \langle 2xy, x^2 \rangle$ conservative? If so, find the potential.

  1. $P_y = 2x = Q_x$ ✓.
  2. $f_x = 2xy \Rightarrow f = x^2y + g(y)$.
  3. $f_y = x^2 + g'(y) = x^2 \Rightarrow g'(y) = 0$.
  4. $f(x,y) = x^2y + C$.
Example 3

Use the potential from Example 2 to evaluate $\int_C \mathbf{F} \cdot d\mathbf{r}$ from $(0,0)$ to $(1,3)$.

  1. $f(1,3) - f(0,0) = 3 - 0 = 3$.

Practice Problems

1. Is $\mathbf{F} = \langle y, -x \rangle$ conservative?
2. Is $\mathbf{F} = \langle 2x, 2y \rangle$ conservative? Find $f$.
3. Evaluate $\int_C xy\,ds$, $C$: $\mathbf{r}(t)=\langle t,t \rangle$, $0 \leq t \leq 1$.
4. $\int_C \mathbf{F}\cdot d\mathbf{r}$: $\mathbf{F}=\langle 1,2 \rangle$, $C$: line from $(0,0)$ to $(3,4)$.
5. Find the work: $\mathbf{F} = \langle x, y \rangle$, path from $(0,0)$ to $(1,1)$ along $y = x$.
6. What does $\int_C \mathbf{F}\cdot d\mathbf{r}$ represent physically?
7. Test: $\mathbf{F} = \langle 3x^2y, x^3+1 \rangle$. Conservative?
8. If $f(x,y) = xy$, evaluate $\int_C \nabla f \cdot d\mathbf{r}$ from $(1,1)$ to $(2,3)$.
9. Parameterize: line from $(1,0)$ to $(0,1)$.
10. $\int_0^{2\pi} \langle -\sin t, \cos t \rangle \cdot \langle -\sin t, \cos t \rangle\,dt$.
11. What is a potential function?
12. If $\mathbf{F}$ is conservative, what is $\oint_C \mathbf{F}\cdot d\mathbf{r}$?
Show Answer Key

1. $P_y = 1$, $Q_x = -1$; not conservative

2. $P_y = 0 = Q_x$; $f = x^2+y^2+C$

3. $\|\mathbf{r}'\| = \sqrt{2}$; $\int_0^1 t^2\sqrt{2}\,dt = \sqrt{2}/3$

4. $\mathbf{r}(t) = \langle 3t,4t \rangle$; $\int_0^1(3+8)\,dt = 11$

5. $\int_0^1(t+t)\,dt = 1$

6. Work done by force $\mathbf{F}$ along path $C$

7. $P_y = 3x^2 = Q_x$; yes, conservative

8. $f(2,3)-f(1,1) = 6-1 = 5$

9. $\mathbf{r}(t) = \langle 1-t, t \rangle$, $0 \leq t \leq 1$

10. $\int_0^{2\pi}(\sin^2 t + \cos^2 t)\,dt = 2\pi$

11. $f$ such that $\nabla f = \mathbf{F}$

12. $0$ (for any closed path)

∮ Line Integral Calculator
∮ F · dr (closed circle)
Conservative?
Green's ∬(∂Q/∂x−∂P/∂y)dA
Gradient & Directional Derivatives Placement Test Practice — Multivariable Calculus