Vector Line Integrals

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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

⟨ 5, 0 ⟩ .

The current pushes the boat in the direction

⟨ 2 \sqrt{2}, 2 \sqrt{2} ⟩ .

(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) .

The current has a constant value of

⟨ 2, 0 ⟩ .

(i)

What direction is the boat heading at

t = 0 ?

t = π / 2 ?

(ii)

How much help is the current at

t = 0 ?

t = π / 2 ?

(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)

\int_{C} \vec{F} \cdot d \vec{r} = \int_{a}^{b} \vec{F} (\vec{r} (t)) \cdot {\vec{r}}^{'} (t) d t .

Exercises Exercises

Evaluate the following vector line integrals.

1.

\vec{F} (x, y) = ⟨ 5 x, 3 y ⟩

along

\vec{r} (t) = ⟨ 2 t, 3 t ⟩

for

t \in [0, 2] .

2.

\vec{F} (x, y) = ⟨ 5 x, 3 y ⟩

along the line from

(0, 0)

to

(5, 1)

3.

\vec{F} (x, y) = ⟨ 5 x, 3 y ⟩

along the line from

(5, 1)

to

(0, 0)

4.

\vec{F} (x, y) = ⟨ 5 x, 3 y ⟩

along the unit circle

5.

\vec{F} (x, y) = ⟨ 5 x, 3 y ⟩

along the triangle with vertices

(0, 0),

(3, 0),

and

(0, 4) .

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