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Chapter 6: Q8P (page 323)

Evaluate each of the integrals in Problems3to 8as either a volume integral or a surface integral, whichever is easier.

鈭�鈭�脳痴诲蟿over the volumerole="math" localid="1657334446941" x2+y2鈮�4,0鈮�z鈮�5,V=(x2+y2)(ix+jy)

Short Answer

Expert verified

The solution of the integrals is 鈭�鈭�脳Vd蟿=80蟺.

Step by step solution

01

Given Information.

The given integral is鈭�鈭�脳Vd蟿.

02

Definition of Divergence’s Theorem.

Divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed. According to this theorem, the surface integral of a vector field over a closed surface, also known as the flux through the surface, equals the volume integral of the divergence over the region inside the surface.

03

Apply Gauss’ Theorem.

Apply Gauss' theorem and use the fact mentioned below.

V脳蚁^=蚁2,

蚁=2

In cylindrical coordinates.d蟽=蚁d胃dz

Solve further as shown below.

鈭�V脳蚁^=鈭�05鈥�鈭�02蟺鈥�23d胃dz

=8(2蟺)(5)

=80蟺

Hence, the solution of the integrals is 鈭�鈭�脳Vd蟿=80蟺.

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Most popular questions from this chapter

For Problem 11,

(a) Find the magnitude and direction of the electric field at (2,1).

(b) Find the direction in which the temperature is decreasing most rapidly at(-3,2)

(c) Find the rate of change of temperature with distance at (1,2)in the direction

鈭�(y2-x2)dx+(2xy+3)dyalong the x axis from (0,0) to(5,0) and along a circular are from(5,0) to (1,2).

Question: 鈭�鈭�curl(x2yi-xzk)路苍诲蟽over the closed surface of the ellipsoid

.x24+y29+z216=1

Warning: Stokes鈥� theorem applies only to an open surface. Hints: Could you cut the given surface into two halves? Also see (d) in the table of vector identities (page 339).

鈭�F.draround the circle over the curved part of the hemisphere in Problem 24, if x2+y2+2x=0, where F=yi-xj.

Find the derivative of xy2+yz at (1,1,2) in the direction of the vector 2i-j+2k.
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