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Chapter 13: Q16E (page 760)

Match the vector field F on \({{\rm{R}}^{\rm{3}}}\)with the plots labelled I–IV. Give reasons for your choices.

\(F(x,y,z) = i + 2j + zk\)

Short Answer

Expert verified

The vector field \(F\) matches with the plot I.

Step by step solution

01

Given Information.

The given vector field is \(F(x,y,z){\rm{ }} = {\rm{ }}i + 2j + zk\).

02

Select the matching plot and give reason.

It is a vector that has constant direction in xy-plane and it has a variable z component.

In figure I, the vectors have constant direction and magnitude in xy-plane and the length in z-direction changes.

Therefore, the vector field \(F\) matches with the plot I.

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Most popular questions from this chapter

Evaluate the line integral \(\int_C {{x^2}} dx + {y^2}dy\) for the curve \(C\) which consists of the arc of the circle \({x^2} + {y^2} = 4\) from the point \((2,0)\) to \((0,2)\) followed by the line segment from the point \((0,2)\) to the point \((4,3)\).

Let \({\rm{F}}\)be the vector field shown in the figure.

(a) If \({{\rm{C}}_{\rm{1}}}\)is the vertical line segment from\({\rm{( - 3, - 3)}}\)to\({\rm{( - 3,3)}}\), determine whether \(\int_{{{\rm{C}}_{\rm{1}}}} {\rm{F}} \cdot {\rm{dr}}\) is positive, negative, or zero.

(b) If \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)is the counterclockwise-oriented circle with a radius of 3 and center the origin, determine whether \(\int_{{{\rm{C}}_{\rm{2}}}} {\rm{F}} \cdot {\rm{dr}}\)is positive, negative, or zero.

Find, to four decimal places, the area of the part of the surface \(z = \frac{{1 + {x^2}}}{{1 + {y^2}}}\), that lies above the square \(\left| x \right| + \left| y \right| \le 1\). Illustrate by graphing this part of the surface.

The curve with the vector equation \({\rm{r(t) = }}{{\rm{t}}^{\rm{3}}}{\rm{i + 2}}{{\rm{t}}^{\rm{3}}}{\rm{j + 3}}{{\rm{t}}^{\rm{3}}}{\rm{k}}\) is a line.

Sketch the vector field by drawing a diagram like Figure 4 or Figure 8.

\(F(x,y) = yi{\rm{ + }}\left( {x{\rm{ + }}y} \right)j\)
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