Calculo

F.
TX.10

17

VECTOR CALCULUS

ET 16

17.1 Vector Fields

ET 16.1

1. F({> | ) = 1 (i + j)
2

All vectors in this ¿eld are identical, with length

1
I
2

and

direction parallel to the line | = {.

3. F({> | ) = | i +

1
2

j

The length of the vector | i +

1
2

j is

t
| 2 + 1 . Vectors
4

are tangent to parabolas opening about the {-axis.

|i + {j
{2 + | 2

5. F({> | ) = s

|i +{j
The length of the vector s is 1.
{2 + | 2

7. F({> |> } ) = k

All vectors in this ¿eld are parallel to the } -axis and have length 1.

9. F({> |> } ) = { k

At each point ({> |> } ), F({> |> } ) is a vector of length |{|.
For { A 0, all point in the direction of the positive } -axis, while for { ? 0, all are in the direction of the negative
} -axis. In each plane { = n, all the vectors are identical.

269

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270

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CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16

TX.10

11. F({> | ) = h|> {i corresponds to graph II. In the ¿rst quadrant all the vectors have positive {- and | -components, in the second

quadrant all vectors have positive {-components and negative |-components, in the third quadrant all vectors have negative {and |-components, and in the fourth quadrant all vectors have negative {-components and positive | -components. In addition, the vectors get shorter as we approach the origin.
13. F({> | ) = h{ 3 2> { + 1i corresponds to graph I since the vectors are independent of | (vectors along vertical lines are

identical) and, as we move to the right, both the {- and the | -components get larger.
15. F({> |> } ) = i + 2 j + 3 k corresponds to graph IV, since all vectors have identical length and direction.
17. F({> |> } ) = { i + | j + 3 k corresponds to graph III; the projection of each vector onto the {| -plane is { i + | j, which points

away from the origin, and the vectors point generally upward because their } -components are all 3.
19.

The vector ¿eld seems to have very short