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Chapter 6: Q3P (page 334)

Use either Stokes' theorem or the divergence theorem to evaluate each of the following integrals in the easiest possible way.

鈭�surface蟽鈥�curl鈦�(x2i+z2j鈭�y2k)鈭�苍诲蟽, where 蟽 is the part of the surface z=4鈭�x2鈭�y2above the (x,y) plane.

Short Answer

Expert verified

The approximation of the integral isln鈦�(n!)=nln鈦�(n)鈭�n

Step by step solution

01

Given Information.

The given integral is鈭�surfiace蟽鈥�curl鈦�x2i+z2j鈭�y2k脳苍诲蟽

02

Definition of Divergence Theorem.

According to the divergence theorem, the surface integral of a vector field over a closed surface, known as the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.

03

Evaluate the Integral.

Use the expansion of a factorial and for demonstration, use only the first two terms.

ln鈦�n!=ln鈦�nne鈭�n2蟺苍1+112n

=ln鈦�nn+ln鈦�e鈭�n+ln鈦�n1/2+ln鈦�1+112n

=nln鈦�n鈭�n+12ln鈦�n+ln鈦�1+112n

Use the value n=1023.

nln鈦�n=5.526脳1026

n=1036

Inn=55ln鈦�1+112n鈮�0

All the subsequent terms are even smaller and contribute lesser. Therefore, the solution of the integral is ln鈦�(n!)=nln鈦�(n)鈭�n.

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Most popular questions from this chapter

鈭�Cexcosydx-exsinydy, whereCis the broken line fromA=(ln2,0)to D=(0,1)and then from DtoB=(-ln2,0).

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).

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The force on a charge moving with velocity v=dr/dtin a magnetic field B isF=q(v脳B)we can write B asB=鈭�脳Awhere A (called the vector potential) is a vector function of x,y,z,t . If the position vectorr=ix+jy+kzof the charge is a function of time, show that

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Thus show that

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