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Chapter 14: Q. 3 (page 1131)

In what sense is the integrand of the double integral in Green鈥檚 Theorem the antiderivative of the vector field?

Short Answer

Expert verified

The terms in the integrand of the double integral are the mixed partial derivatives of the component functions of the vector field Fx,y.

Step by step solution

01

Step 1. Given information.

The objective is to find in what sense the integrand of the double integral in Green's Theorem is the antiderivative of the vector field.

02

Step 2. Green's Theorem

Green's Theorem states that,
"Let R be a region in the plane with a smooth boundary curve C oriented counterclockwise by r(t)=鉄�(x(t),y(t))鉄�fora鈮�t鈮�b

If a vector field F(x,y)=F1(x,y),F2(x,y)is defined on R, then,

鈭�C鈥�F鈰�dr=鈭�R鈥�鈭�F2鈭�x鈭�鈭�F1鈭�ydA."

03

Step 3. Explanation

Consider the double integral in Green's Theorem: 鈭�R鈥�鈭�F2鈭�x鈭�鈭�F1鈭�ydA

Notice that, the terms in the integrand of the double integral are the mixed partial derivatives of the component functions of the vector field F(x,y).

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Most popular questions from this chapter

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.F(x,y)=yx2i鈭�xcosyj

How would you show that a given vector field in 鈩�2 is not conservative?

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鈥� 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鈥� Theorem can be interpreted as a generalization of Green鈥檚 Theorem.

(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.

(f) True or False: If F(x, y ,z) is a conservative vector field, then Stokes鈥� 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 .

Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.

Problem Zero: Read the section and make your own summary of the material.
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