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Chapter 6: Q22P (page 335)

Find vector fieldsA such that V=curlAfor each given V.V=(y+z)i+(x鈭�z)j+x2+y2k

Short Answer

Expert verified

The vector field derived isA=ixz鈭�y33+j鈭�yz+x33+k(x+y)z+鈭�u.

Step by step solution

01

Given Information.

The given equation isV=(y+z)i+(x鈭�z)j+x2+y2k.

02

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

03

Find the vector field.

In this problem find vector A, whose curl gives the vector

V=(y+z)i+(x鈭�z)j+x2+y2k

The components of vector are given in terms of the derivatives of A.

鈭�Az鈭�y鈭�鈭�Ay鈭�z=y+z

鈭�Ax鈭�y鈭�鈭�Az鈭�z=x-z

鈭�Ay鈭�x鈭�鈭�Ax鈭�y=x2+y2

Observe the first two equation Az=Z(x+y), and choose the equation.

The first equation then gives the equation 鈭�鈭�Ay鈭�z=y.

Integrate to obtain the equation Ay=鈭�yz+f(x,y).

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Most popular questions from this chapter

(a) Suppose that a hill (as in Fig. 5.1) has the equation 32-x2-4y2, where z=heightmeasuredfromsomerefrencelevel(in hundreds of feet). Sketch acontour map (that is, draw on one graph a set of curvesz=const.); use the contours z=32,19,12,7,0(b) If you start at the point(3,2)and in the directioni+j, are you going up hillor downhill, and how fast?

Find vector fields A such that V = curl A for each given V.

V=i(x2-2xz)+j(y2-2xy)+k(z2-2yz+xy)

Evaluate鈭�V鈰�nd蟽over the curved surface of the hemispherex2+y2+z2=9,z鈮�0, ifV=yi+xzj+(2z-1)k.Careful: See Problem 9.

Show that 鈭�.(U脳r)=r.(鈭�脳U)where U is a vector function of x,y,zandr=xi+yj+zk .

鈭�(y2-x2)dx+(2xy+3)dyalong the x axis from (0,0) to(5,0) and along a circular are from(5,0) to (1,2).
See all solutions

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