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Chapter 14: Q. 34 (page 1096)

Show that the vector fields in Exercises 33–40 are not conservative.

F(x,y)=(x2+y2,cosy)

Short Answer

Expert verified

The vector fields is not conservative because ∂F1∂y≠∂F2∂x.

Step by step solution

01

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
F(x,y)=(x2+y2,cosy)

02

Step 2. A vector field F(x,y)=(F1(x,y),F2(x,y)) is not conservative if and only if ∂F1∂y≠∂F2∂x.

For the vector fieldF(x,y)=(x2+y2,cosy)

∂F1∂y=∂∂y(x2+y2)∂F1∂y=∂∂yx2+∂∂yy2∂F1∂y=0+2y∂F1∂y=2y

03

Step 3. Now finding ∂F2∂y

∂F2∂x=∂∂xcosy∂F2∂x=0

Hence, ∂F1∂y≠∂F2∂x.

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Most popular questions from this chapter

If the velocity of a flow of a gas at a point (x, y, z) is represented by F and the gas is expanding at that point, what does this imply about the divergence of F at the point?

What is the difference between the graphs of

G(x,y,z)=2i−3j+zkandF(x,y,z)=−2i+3j−zk

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing ∫Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then ∫Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then ∫SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing ∫SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing ∫SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.

Q. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Stokes’ Theorem asserts that the flux of a vector field through a smooth surface with a smooth boundary is equal to the line integral of this field about the boundary of the surface.

(b) True or False: Stokes’ Theorem can be interpreted as a generalization of Green’s Theorem.

(c) True or False: Stokes’ Theorem applies only to conservative vector fields.

(d) True or False: Stokes’ Theorem is always used as a way to evaluate difficult surface integrals.

(e) True or False: Stokes’ Theorem can be interpreted as a generalization of the Fundamental Theorem of Line Integrals.

(f) True or False: If F(x, y ,z) is a conservative vector field, then Stokes’ Theorem and Theorem 14.12 together give an alternative proof of the Fundamental Theorem of Line Integrals for simple closed curves.

(g) True or False: Stokes’ Theorem can be interpreted as a generalization of the Fundamental Theorem of Calculus.

(h) True or False: Stokes’ Theorem can be used to evaluate surface area .

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).
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