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Lessons for Multivariable Calculus

Mark Fitch

Section 5.3 Vector Field Theorems

The following sections will develop properties of nice vector fields and how that affects vector line integrals.

Subsection 5.3.1 Path Independence

Activity 70.

The purpose of this activity is to compare the results of integrating along different paths between the same two points for a nice vector field.
For the following problems use \(\vec{F}(x,y)=\langle 2x,2y \rangle.\) The vector field and curves are shown in Figure 5.3.1.
(a)
Calculate the vector line integral of \(\vec{F}\) along each of the following curves.
(i)
\(C_1\text{:}\) \(x=t,\) \(y=t,\) \(t \in [0,1].\)
(ii)
\(C_2\text{:}\) \(x=t,\) \(y=t^2,\) \(t \in [0,1].\)
(iii)
\(C_3\text{:}\) \(x=t,\) \(y=t^3,\) \(t \in [0,1].\)
(iv)
\(C_4\text{:}\) \(x=1-\cos t,\) \(y=\sin t,\) \(t \in [0,\pi/2].\)
(b)
Using the graphs determine list the curves in order of length.
(c)
Do the integral values calculated above vary with the length of the curve?
(d)
Using the graphs explain this result.
Figure 5.3.1. Integration Over Different Domains

Subsection 5.3.2 Fundamental Theorem of Vector Line Integrals

Activity 71.

The purpose of this activity is to recognize a property of scalar integrals in these vector line integrals over nice vector fields.
For the following problems use \(F(x,y)=x^2+y^2.\)
(a)
Calculate \(\nabla F.\)
(b)
What function was integrated in the line integrals in Activity 70 (be general)?
(c)
Calculate \(F(x,y)\) at the beginning and end of the following curves.
(i)
\(C_1\text{:}\) \(x=t,\) \(y=t,\) \(t \in [0,1].\)
(ii)
\(C_2\text{:}\) \(x=t,\) \(y=t^2,\) \(t \in [0,1].\)
(iii)
\(C_3\text{:}\) \(x=t,\) \(y=t^3,\) \(t \in [0,1].\)
(iv)
\(C_4\text{:}\) \(x=1-\cos t,\) \(y=\sin t,\) \(t \in [0,\pi/2].\)
(d)
Compare the results of the line integrals in Activity 70 to the values of \(F\) you just calculated.
(e)
Where did you see this in Calculus 1?

Subsection 5.3.3 Closed Curves

Activity 72.

The purpose of this activity is to extend the result of the previous activity to a special case (type of curve).
For the following problems use \(\vec{F}(x,y)=\langle 2x,2y \rangle\) and the curves from the sections above.
(a)
Calculate without the Fundamental Theorem of Line Integrals the vector line integrals of \(\vec{F}\) over the following curves.
(i)
\(C_4\) but change to \(t \in [0,2\pi]\text{.}\)
(ii)
Out \(C_1\) and back by \(C_2\text{.}\)
(iii)
Out \(C_2\) and back by \(C_3\text{.}\)
(iv)
\(C_5\text{:}\) \(x=5\cos t\text{,}\) \(y=3\sin t\text{,}\) \(t \in [0,2\pi]\text{.}\)
(b)
What is the result?
(d)
If the Fundamental Theorem of Line Integrals applies, would it have predicted this result? Explain.

Subsection 5.3.4 Conservative Fields

The nice fields are called conservative fields because of the property demonstrated in Subsection 5.3.3

Definition 5.3.2. Conservative Field.

A field \(\vec{F}\) is conservative if and only if \(\vec{F}=\nabla f\) for some function \(f\text{.}\)

Activity 73.

The purpose of this activity is to develop a test to determine if a field is conservative.
(a)
For each of the following fields calculate the vector line integral of the given field over \(C_1: (t,1-t)\) for \(t \in [0,1]\) and \(C_2: (\cos t, \sin t)\) for \(t \in [0,\pi/2]\text{.}\) Do not use the Fundamental Theorem of Line Integrals as it does not apply to some of the fields. Technology may be used for the calculations.
(i)
\(\vec{F}(x,y)=\langle y,x \rangle\text{.}\)
(ii)
\(\vec{G}(x,y)=\langle y,-x \rangle\text{.}\)
(iii)
\(\vec{H}(x,y)=\langle 2x-y,2y-x \rangle\text{.}\)
(iv)
\(\vec{J}(x,y)=\langle 2x-y,2y \rangle\text{.}\)
(b)
For each of the functions above check if it is the gradient of some function. (Hint: calculate \(\int \vec{F}_x \; dx \) and \(\int \vec{F}_y \; dy \) and compare.)
(c)
From Theorem 2.2.3 what is true for nice functions \(f(x,y)\) about \(\frac{\partial^2 f}{\partial x \partial y}\) and \(\frac{\partial^2 f}{\partial y \partial x}\text{?}\)
(d)
Because a conservative field is a gradient, what should be true about \(\frac{\partial \vec{F}_x}{\partial y}\) and \(\frac{\partial \vec{F}_y}{\partial x}\text{.}\)

Exercises 5.3.5 Exercises

Integrate each function along the specified curve. Use the most efficient technique.

1.

\(\vec{f}(x,y)=\langle x-y,x-2 \rangle\) along the unit circle.

2.

\(\vec{g}(x,y)=\langle \cos y,-x\sin y \rangle\) along the triangle with vertices \((0,0),\) \((5,0),\) \((6,0).\)

3.

\(\vec{h}(x,y)=\langle |x|,|y| \rangle\) along the line \(y=x\) for \(x \in [-1,1].\)

4.

\(\vec{j}(x,y,z)=\langle 2x+yz,2y+xz,2z+xy \rangle\) along the curve \((t,\cos(\pi t/5),e^{3t})\) for \(t \in [0,7]. \)
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