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Chapter 13: Q7E (page 760)

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

\(F(x,y,z) = k\)

Short Answer

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Step-by-Step Solution

Step by step solution

01

Concept

To sketch a vector field, draw the arrow representing the vector\({\rm{F}}\left( {{\rm{x,y,z}}} \right){\rm{ = k}}\)starting at point\(\left( {{\rm{x, y, z}}} \right)\).

It is impossible to do this for all points\(\left( {{\rm{x, y, z}}} \right)\)but we can gain a reasonable impression of\({\rm{F}}\)by doing it for a few representative points in the domain.

02

Given Information.

The given vector field is \(F\left( {x,y,z} \right) = k\).

03

Calculation.

Let us rewrite the vector field as \(F\left( {0,0,1} \right)\)

The vector field is independent of variables \(x,y,z\).

The \(z\) component is \(1\) (positive component)

Thus, all vectors are vertical and pointing upward direction, with unit magnitude.

04

Sketch the vector field.

The sketch of the vector field will be:

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Most popular questions from this chapter

To find: The area of the surface that creates by the intersection of two cylinders \({y^2} + {z^2} = 1\) and \({x^2} + {z^2} = 1\).

Find the value of \(\iint_S ydS\)\(z = \frac{2}{3}\left( {{x^{\frac{3}{2}}} + {y^{\frac{3}{2}}}} \right),0 \le x \le 1\)and \(0 \le y \le 1.\)

(a) Find a function \(f\) such that \({\bf{F}} = \nabla f\) and (b) use part (a) to evaluate \(\int_C {\bf{F}} \cdot d{\bf{r}}\) along the given curve \(C\).

\({\bf{F}}(x,y,z) = \left( {{y^2}z + 2x{z^2}} \right){\bf{i}} + 2xyz{\bf{j}} + \left( {x{y^2} + 2{x^2}z} \right){\bf{k}}\),

Show that the line integral is independent of the path and evaluate the integral.

\(\int_C {\sin } ydx + (x\cos y - \sin y)dy\),

\(C\)is any path from \((2,0)\) to \((1,\pi )\)

If you have a CAS that plots vector fields (the command is field plot in Maple and Plot Vector Field or Vector Plot in Mathematica), use it to plot\({\bf{F}}(x,y) = \left( {{y^2} - 2xy} \right){\bf{i}} + \left( {3xy - 6{x^2}} \right){\bf{j}}\)

Explain the appearance by finding the set of points \((x,y)\) such that \({\bf{F}}(x,y) = {\bf{0}}\).
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