Chapter 16.9, Problem 11E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫_{s}**F** · *d*S; that is, calculate the flux of **F** across *S.*

**11. F**(*x, y, z*) = (2*x*^{3} + *y*^{3}) **i** + (*y*^{3} + *z*^{3})j + 3*y*^{2}*z* **k**, *S* is the surface of the solid bounded by the paraboloid *z =* 1 - *x*^{2} - *y*^{2} and the *xy*-plane

To determine

**To calculate:** The flux of the vector field
*S*, which is bounded by the paraboloid
*xy*-plane.

Explanation

**Given data:**

The vector field is

The surface of solid
*xy*-plane.

**Formula used:**

Write the expression to find flux of the vector field

Here,

Write the expression to find divergence of vector field

Write the equation of paraboloid as follows.

From the equation of paraboloid, the parameters are considered as follows.

Substitute

As the surface
*xy*-plane, the limits of *z*, *r*, and

**Calculation of**
**:**

Substitute
*P*,
*Q*, and
*R* in equation (2),

Substitute

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P-49ESect-16.2 P-50ESect-16.2 P-51ESect-16.2 P-52ESect-16.3 P-1ESect-16.3 P-2ESect-16.3 P-3ESect-16.3 P-4ESect-16.3 P-5ESect-16.3 P-6ESect-16.3 P-7ESect-16.3 P-8ESect-16.3 P-9ESect-16.3 P-10ESect-16.3 P-11ESect-16.3 P-12ESect-16.3 P-13ESect-16.3 P-14ESect-16.3 P-15ESect-16.3 P-16ESect-16.3 P-17ESect-16.3 P-18ESect-16.3 P-19ESect-16.3 P-20ESect-16.3 P-21ESect-16.3 P-22ESect-16.3 P-23ESect-16.3 P-24ESect-16.3 P-25ESect-16.3 P-26ESect-16.3 P-28ESect-16.3 P-29ESect-16.3 P-30ESect-16.3 P-31ESect-16.3 P-32ESect-16.3 P-33ESect-16.3 P-34ESect-16.3 P-35ESect-16.4 P-1ESect-16.4 P-2ESect-16.4 P-3ESect-16.4 P-4ESect-16.4 P-5ESect-16.4 P-6ESect-16.4 P-7ESect-16.4 P-8ESect-16.4 P-9ESect-16.4 P-10ESect-16.4 P-11ESect-16.4 P-12ESect-16.4 P-13ESect-16.4 P-14ESect-16.4 P-17ESect-16.4 P-18ESect-16.4 P-19ESect-16.4 P-20ESect-16.4 P-21ESect-16.4 P-22ESect-16.4 P-23ESect-16.4 P-24ESect-16.4 P-25ESect-16.4 P-26ESect-16.4 P-27ESect-16.4 P-28ESect-16.4 P-29ESect-16.4 P-30ESect-16.4 P-31ESect-16.5 P-1ESect-16.5 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