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Chapter 6: Q21P (page 336)

Find vector fields A such that V=curlAfor each given V.

V=-K

Short Answer

Expert verified

The vector field is derived to beA=iy+鈭�u .

Step by step solution

01

Given Information.

The given equation isV=-k .

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 it magnitude.

03

Find the vector field.

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

V=-K

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

role="math" localid="1659242905787" 鈭�Az鈭�y-鈭�Ay鈭�z=0鈭�Ax鈭�z-鈭�Az鈭�x=0鈭�Az鈭�y-鈭�Ay鈭�z=0鈭�Ay鈭�x-鈭�Ax鈭�y=-1

An obvious solution is mentioned below.

A=iy+鈭�u

Here is an arbitrary function.

Hence, the vector field is derived.

A=iy+鈭�u

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Most popular questions from this chapter

For a simple closed curve Cin the plane show by Green鈥檚 theorem that the area inclosed is

A=12鈭�C(xdy-ydx)

Show by the Lagrange multiplier method that the maximum value of 诲蠄/dsis|诲蠄|.That is, maximize 诲蠄/dsgiven by (6.3) subject to the condition a2+b2+c2=1. You should get two values (卤) for the Lagrange multiplier 位, and two values (maximum and minimum) for诲蠄/dswhich is the maximum and which is the minimum?

Find the derivative of zexcosy at (1,0,蟺3) in the direction of the vector i+2j .

Givenu=xy+yz+zsinx, find

(a) 鈭�uat(0,1,2);

(b) The directional derivative of (0,1,2) at in the direction 2i+2j-k;

(c) The equations of the tangent plane and of the normal line to the level surface u=2at(0,1,2);

(d) a unit vector in the direction of most rapid increase of u at(0,1,2)

Suppose the density 蚁 varies from point to point as well as with time, that is, 蚁=(x,y,z,t). If we follow the fluid along a streamline, then x,y,z are function of such that the fluid velocity is

v=idxdt+jdydt+kdzdt

Show that thend蚁/dt=鈭�蚁/鈭�t+v鈭�鈭�蚁 . Combine this equation with (10.9)to get

蚁鈭�鈭�v+d蚁dt=0

(Physically, is the rate of change of density with time as we follow the fluid along a streamline; 鈭�p/鈭�tis the corresponding rate at a fixed point.) For a steady state (that is, time-independent), 鈭�p/鈭�t=0, but 鈭�p/鈭�t is not necessarily zero. For an incompressible fluid, 鈭�p/鈭�t=0. Show that then role="math" localid="1657336080397" 鈭�脳v=0. (Note that incompressible does not necessarily mean constant density since 鈭�p/鈭�t=0does not imply either time or space independence of 蚁; consider, for example, a flow of watermixed with blobs of oil.)
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