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Chapter 14: Q. 16 (page 1119)

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.

Short Answer

Expert verified

Hence,theaveragerateoffluxofavectorfieldf(x,y,z)onasmoothsurfaceSis,Fav(S)=鈭�S鈥�F(x,y,z)鈰�ndS鈭�S鈥�1dS

Step by step solution

01

Step 1. Given

The given function is f(x, y, z) .

02

Step 2. Finding the average rate of flux of a vector field through smooth surface

Surfaceintegralsofmultivariatefunctionsf(x,y,z)is,鈭�f(x,y,z)n.dS.ThesurfaceareaofthesmoothsurfaceSis,鈭�S1dSHence,theaveragerateoffluxofavectorfieldf(x,y,z)onasmoothsurfaceSis,Fav(S)=鈭�S鈥�F(x,y,z)鈰�ndS鈭�S鈥�1dS

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Most popular questions from this chapter

Evaluate the integrals in Exercises 43鈥�46 directly or using Green鈥檚 Theorem.

鈭�R3xy-4x2ydA, where R is the unit disk.

Suppose that an electric field is given by

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Compute the flux 鈭�S鈥�E鈰�ndA of the field through the unit cube [0,1]脳[0,1]脳[0,1].

S is the portion of the saddle surface determined by z = x2 鈭� y2 that lies above and/or below the annulus in the xy-plane determined by the circles with radii

32and2

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F(x,y,z)=鉄�yz,xz,xy鉄�, where S is the portion of the saddle determined byz=x2鈭�y2 that lies above the region in thexy-plane bounded by the x-axis and the parabola with equationy=1鈭�x2.

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.F(x,y,z)=sin-1(xy)i+ln(y+z)j+12x+3y+5z+1k
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