Skip to main content

Active Calculus - Multivariable

Nicholas Long, Mitchel T. Keller, Steve Schlicker

Print preview

Paper size

   

Activity 13.2.4.
Figure 13.2.15 shows a vector field \(\vF\) as well as six oriented curves, as labeled in the plot.
🔗
described in detail following the image
A vector field in the first quadrant with \(x,y\leq 5\text{.}\) Vectors are parallel to the \(y\)-axis and point in the negative \(y\)-direction. Vectors get longer as distance from the \(y\)-axis increases. There are six labeled oriented curves. The curve \(C_1\) is the line segment from \((3,3)\) to \((4,3)\text{.}\) The curve \(C_2\) is the line segment from \((4,3)\) to \((4,5)\text{.}\) The curve \(C_3\) is the line segment from \((4,5)\) to \((3,5)\text{.}\) The curve \(C_4\) is the line segment from \((3,5)\) to \((3,3)\text{.}\) The curve \(C_5\) is the lower half of the circle of radius \(1\) centered at \((3,2)\) oriented counterclockwise. The curve \(C_6\) is the line segment from \((1,1)\) to \((1,5)\text{.}\)
Figure 13.2.15. A vector field \(\vF\) and six oriented curves.
🔗
(a)
Is \(\int_{C_6}\vF\cdot d\vr\) positive, negative, or zero? Explain.
🔗
🔗
(b)
Let \(C = C_1+C_2+C_3+C_4\text{.}\) Determine if \(\displaystyle\int_C\vF\cdot d\vr\) is positive, negative, or zero.
🔗
🔗
(c)
Order the line integrals below from smallest to largest.
\begin{equation*} \int_{C_1}\vF\cdot d\vr\quad \int_{C_2}\vF\cdot d\vr\quad \int_{C_3}\vF\cdot d\vr\quad \int_{C_4}\vF\cdot d\vr\quad \int_{C_5}\vF\cdot d\vr \end{equation*}
🔗
🔗
🔗
🔗
PrevTopNext