Training Multivariable Calculus Vector Fields & Line Integrals
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Vector Fields & Line Integrals

26 min Multivariable Calculus

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 $\mathbf{r}(t) = \langle t, t^2 \rangle$, $0 \leq t \leq 1$.

$\mathbf{r}'(t) = \langle 1, 2t \rangle$. $\mathbf{F}(\mathbf{r}(t)) = \langle t^2, t \rangle$.

$\int_0^1 (t^2 \cdot 1 + t \cdot 2t)\,dt = \int_0^1 3t^2\,dt = 1$.

Example 2

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

$P_y = 2x = Q_x$ ✓. $f_x = 2xy \Rightarrow f = x^2y + g(y)$. $f_y = x^2 + g'(y) = x^2 \Rightarrow g'(y) = 0$.

$f(x,y) = x^2y + C$.

Example 3

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

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

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)

Gradient & Directional Derivatives Placement Test Practice — Multivariable Calculus