   Chapter 16.9, Problem 18E

Chapter
Section
Textbook Problem

Let F(x, y, z) = z tan-1(y2) i + z3 ln(x2 + 1) j + z k. Find the flux of F across the part of the paraboloid x2 + y2 + z = 2 that lies above the plane z = 1 and is oriented upward.

To determine

To find: The flux of the vector field F(x,y,z)=ztan1(y2)i+z3ln(x2+1)j+zk across the part of the paraboloid x2+y2+z=2 .

Explanation

Given data:

The vector field is F(x,y,z)=ztan1(y2)i+z3ln(x2+1)j+zk .

The surface S is the part of the paraboloid x2+y2+z=2 that lies above the plane z=1 and is oriented upward.

Formula used:

Write the expression to find the flux of the vector field.

SFdS=DFndA (1)

Here,

n is the normal vector to the surface and

D is the region of the surface.

Write another expression to find the flux of the vector field.

SFdS=EdivFdV (2)

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 (3)

From the given data, the surface is not a closed surface. Therefore the flux is evaluated by considering the surface as follows.

Consider the surface S1 is the disk x2+y2=1 and the surface S2 is SS1 .

From the given data, the flux of the vector field is determined across the surface S1 with formula in equation (1) and it is determined across the surface S2 with formula in equation (2).

Calculation of S1FdS :

As the surface S1 oriented downward, the unit normal vector to surface is (k) .

n=k

Rewrite the normal vector as follows.

n=(0)i+(0)j+(1)k

Substitute ztan1(y2)i+z3ln(x2+1)j+zk for F and (0)i+(0)j+(1)k for n in equation (1),

S1FdS=D[ztan1(y2)i+z3ln(x2+1)j+zk][(0)i+(0)j+(1)k]dA=D[ztan1(y2)(0)+z3ln(x2+1)(0)+z(1)]dA=D(z)dA=D(z)dA

As the z-component is 1 for the surface S1 disk, substitute 1 for z.

S1FdS=D(1)dA=A(D)=π

As the surface S2 is closed surface, the flux across the surface S2 is determined with the divergence theorem (formula in equation (2)).

Calculation of divF :

Substitute ztan1(y2) for P , z3ln(x2+1) for Q , and z for R in equation (3),

divF=x[ztan1(y2)]+y[z3ln(x2+1)]+z(z)=0+0+1=1

Calculation of S2FdS :

Substitute 1 for divF in equation (2),

S2FdS=E(1)dV (4)

Write the equation of paraboloid as follows

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