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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use Stokes’ Theorem to evaluate ∫c F · dr. In each Case C is oriented counterclockwise as viewed from above.

10. F(x, y, z) = 2y i + xz j + (x + y) k, C is the curve of intersection of the plane z = y + 2 and the cylinder x2 + y2 = l

To determine

To evaluate: The value of CFdr by the use of Stokes’ theorem.

Explanation

Given data:

The field is F(x,y,z)=2yi+xzj+(x+y)k and C is curve of intersection of plane z=y+2 and cylinder x2+y2=1 .

Formula used:

Write the expression for curl of F(x,y,z)=Pi+Qj+Rk .

curlF=|ijkxyzPQR|

curlF=(RyQz)i(RxPz)j+(QxPy)k (1)

Write the expression for the Stokes’ theorem.

CFdr=ScurlFdS (2)

Here,

S is the surface.

Consider surface S, z=g(x,y) is in upward orientation. Write the expression for surface integral of F over surface S.

ScurlFdS=D(PgxQgy+R)dA (3)

Here,

A is the area.

Find the value of curlF by using equation (1).

curlF=((x+y)y(xz)z)i((x+y)x(2y)z)j+((xz)x(2y)y)k=((0+1)x(1))i((1+0)0)j+(z(1)2(1))k=(1x)ij+(z2)k

The surface S is a region which is enclosed by plane C. The expression for surface S, D={(r,θ)|0x1,0y2π} in polar coordinate system is,

z=y+2

Here,

x=rcosθandy=rsinθ .

Hence the equation is in the form of z=g(x,y) .

The curve C is oriented in counter-clockwise, so orient the surface S in upwards.

Compare the equations curlF=Pi+Qj+Rk and curlF=(1x)ij+(z2)k .

P=1xQ=1R=z2

Find the value of ScurlFdS by using equation (3)

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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