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Chapter 14: Q. 3 (page 1131)

In what sense is the integrand of the double integral in Green’s 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)=⟨(x(t),y(t))⟩fora≤t≤b

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

∫C F⋅dr=∬R ∂F2∂x−∂F1∂ydA."

03

Step 3. Explanation

Consider the double integral in Green's Theorem: ∬R ∂F2∂x−∂F1∂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

Integrate the given function over the accompanying surface in Exercises 27–34.
f(x,y,z)=yx4z2+1, where S is the portion of the paraboloid z=x2+y2that lies above the rectangle determined by 1≤x≤eand 0≤y≤2in the xyplane.

Given a smooth parametrization for a “generalized cylinder” S, given by extending the curve y = x2 upwards and downwards from z =−2 to z = 3.

Find the integral of f(x,y,z)=x-y-zon the portion of the plane with the equation

10x−33y+36z=30with 2 ≤ x ≤ 7 and 1 ≤ z ≤ 2.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Two different surfaces with the same area. (Try to make these very different, not just shifted copies of each other.)

(b) Let S be the surface parametrized by r(u,z)=x(u,z)i+y(u,v)j+z(u,v)k.

Give two different unit normal vectors to S at the point r(u0,v0).

(c) A smooth surface that can be smoothly parametrized as r(x,z)=x,f(x),z.

Find the area of S is the portion of the plane with equation x = y + z that lies above the region in the xy-plane that is bounded by y = x, y = 5, y = 10, and the y-axis.
See all solutions

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