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Lessons for Multivariable Calculus

Mark Fitch

Section 5.2 Vector Line Integrals

Section 3.2 presented integrals of point valued functions along a fixed curve. Subsection 5.1.2 presented the effect of a vector field on an inactive object. This section develops the effect of a vector field on an object traveling a known curve not determined by the field.

Subsection 5.2.1 Derivation

Activity 69.

In this activity we consider how calculations from Chapter 1 enable us to calculate the effect of a vector field on a path.
(a)
Guido guides a boat in a river in the direction \(\langle 5,0 \rangle\text{.}\) The current pushes the boat in the direction \(\langle 2\sqrt{2},2\sqrt{2} \rangle\text{.}\)
(i)
Sketch these vectors.
(ii)
Calculate the portion of the current that is beneficial to Guido. It is only beneficial if it helps him travel his intended direction.
(b)
Guido guides a boat in a river along the path \((t,\cos t)\text{.}\) The current has a constant value of \(\langle 2,0 \rangle\text{.}\)
(i)
What direction is the boat heading at \(t=0\text{?}\) \(t=\pi/2\text{?}\)
(ii)
How much help is the current at \(t=0\text{?}\) \(t=\pi/2\text{?}\)
(c)
Under what circumstance will a vector field most help/hinder motion along a path?

Subsection 5.2.2 Evaluation

Definition 5.2.1. Vector Line Integral.

For a continuous vector field \(\vec{F}\) and a smooth curve \(C\) given in parametric by \(C=\vec{r}(t)\)
\begin{equation*} \int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}^\prime(t) \; dt. \end{equation*}

Exercises Exercises

Evaluate the following vector line integrals.
1.
\(\vec{F}(x,y)=\langle 5x,3y \rangle\) along \(\vec{r}(t)=\langle 2t,3t \rangle\) for \(t \in [0,2]\text{.}\)
2.
\(\vec{F}(x,y)=\langle 5x,3y \rangle\) along the line from \((0,0)\) to \((5,1)\)
3.
\(\vec{F}(x,y)=\langle 5x,3y \rangle\) along the line from \((5,1)\) to \((0,0)\)
4.
\(\vec{F}(x,y)=\langle 5x,3y \rangle\) along the unit circle
5.
\(\vec{F}(x,y)=\langle 5x,3y \rangle\) along the triangle with vertices \((0,0)\text{,}\) \((3,0)\text{,}\) and \((0,4).\)
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