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Chapter 6: Q3P (page 314)

xydx+x2dywhere C is as selected.

Short Answer

Expert verified

The integral will give the solution to be 143.

Step by step solution

01

Given information

The given equation is xydx+x2dy,.

02

Definition of conservative force and scalar potential

A force is said to be conservative if 鈭�脳F=0..

The scalar potential is independent of the path. The scalar potential is the sum of potential in all the 3 dimensions calculated separately.

The formula for the scalar potential is W=鈭�F.dr.

03

Use Greens Theorem

Takethe values as mentioned below.

P=xyQ=x2

Use the Green theorem.

鈭�xydx+x2dy,鈭�xdydx鈭�14鈭�0xxdydx鈭�14xdx163-23143

Hence the integral will give the solution to be 143.

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Most popular questions from this chapter

A vector force with components (1,2,3)acts at the point(3,2,1). Find the vector torque about the origin due to this force and find the torque about each of the coordinate axes.

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.)

The force F = i - 2jacts at the point (0, 1, 2) Find the torque of F about the line =(2i-j)t.

Is F = yi+xzj+zk conservative? Evaluate 鈭�F.drfrom along the paths

(a) broken line (0,0,0)to (1,1,1) to (1,1,0) to (1,1,1)

(b) Straight line connecting the points.

Evaluate the line integral 鈭�cy2dx+2xdy+dx where Cconnects (0,0,0)with(1,1,1,)

(a) Along straight lines from (0,0,0,)to(1,0,0)to(1,0,1)to(1,1,1,);

(b) on the circle x2+y2-2y=0to(1,1,0)and then on a vertical line to(1,1,1).
See all solutions

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