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Chapter 14: Q. 12 (page 1095)

What is the difference between the graphs of

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

Short Answer

Expert verified

The difference between the graphs are F(x,y,z)=-G(x,y,z);at(0,0).

Step by step solution

01

Step 1.Given Information

What is the difference between the graphs of

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

02

Step 2. The given vector is G(x,y,z)=2i−3j+zk and F(x,y,z)=−2i+3j−zk.

Foreverypoint(x,y,z)鈭�鈩�3WecanwriteF(x,y)asF(x,y,z)=鈭�2i+3j鈭�zkF(x,y,z)=鈭�(2i-3j+zk)AsweknowG(x,y,z)=2i鈭�3j+zkSowecanwriteF(x,y,z)=-G(x,y,z)HenceF(x,y,z)=-G(x,y,z);at(0,0)

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Most popular questions from this chapter

Give a smooth parametrization of the upper half of the unit sphere in terms of x and y.

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.

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

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

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

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

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

Fx,y,z=cosxyzi+j-yzk, where S is the portion of the surface with equation z=y3-y2that lies above and/or below the rectangle determined by 鈭�3鈮�x鈮�2and 鈭�1鈮�y鈮�1 in the xy-plane, with n pointing in the positive z direction.

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.F(x,y,z)=sin-1(xy)i+ln(y+z)j+12x+3y+5z+1k

Use the curl form of Green鈥檚 Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.
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