   Chapter 16.9, Problem 10E

Chapter
Section
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫s F · dS; that is, calculate the flux of F across S.10. F(x, y, z) = z i + y j + zx k, S is the surface of the tetrahedron enclosed by the coordinate planes and the plane x a   +   y b   +   z c   =   1 where a, b, c and c are positive numbers

To determine

To calculate: The flux of vector field F(x,y,z)=zi+yj+zxk across the surface S .

Explanation

Given data:

The vector field is F(x,y,z)=zi+yj+zxk .

The surface S is the tetrahedron that enclosed by the coordinate planes and the plane xa+yb+zc=1 , where a , b , and c are the positive numbers.

Formula used:

Write the expression to find flux of the vector field F(x,y,z) across the surface S .

SFdS=EdivFdV (1)

Here,

E is the solid region.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (2)

Write the equation of tetrahedron as follows.

xa+yb+zc=1

From the equation of tetrahedron, the tetrahedron has the vertices. (0,0,0) , (a,0,0) , (0,b,0) , and (0,0,c) .

From the tetrahedron, the limits of coordinates are written as follows.

0xa0yb(1xa)0zc(1xayb)

Calculation of divF :

Substitute z for P , y for Q , and zx for R in equation (2),

divF=x(z)+y(y)+z(zx)=0+1+x=x+1

Calculation of flux of vector field:

Substitute (x+1) for divF in equation (1),

SFdS=E(x+1)dV

Apply the limits of coordinates and rewrite the expression as follows.

SFdS=0a0b(1xa)0c(1xayb)(x+1)dxdydz=0a0b(1xa)(x+1)dxdy0c(1xayb)(1)dz=0a0b(1xa)(x+1)dxdy[z]0c(1xayb)=0a0b(1xa)(x+1)dxdy[c(1xayb)0]

Simplify the expression as follows

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