Chapter 16.8, Problem 9E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use Stokes’ Theorem to evaluate ∫_{c}**F** · *d*r*.* In each Case *C* is oriented counterclockwise as viewed from above.

**9. F**(*x*, *y*, *z*) = *xy* **i*** + yz* **j** + *zx* **k**, *C* is the boundary of the part of the paraboloid *z* = 1 - *x*^{2} *- y*^{2} in the first octant

To determine

**To evaluate**: The value of

Explanation

**Given data:**

The field is
*C* is boundary of part of paraboloid

**Formula Used:**

Write the expression for curl of

Write the expression for the Stokesâ€™ theorem.

Here,

*S* is the surface.

Consider surface *S*,
*S*.

Here,

*A* is area.

Find the value of

The surface *S* is a paraboloid region which is enclosed by plane *C*. The expression for surface *S*,

Here,

Hence the equation is in the form of

The curve *C* is oriented in counter-clockwise, so the orient the surface *S* upward.

Compare the equations

Find the value of

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P-7ESect-16.8 P-8ESect-16.8 P-9ESect-16.8 P-10ESect-16.8 P-11ESect-16.8 P-12ESect-16.8 P-13ESect-16.8 P-14ESect-16.8 P-15ESect-16.8 P-17ESect-16.8 P-18ESect-16.8 P-19ESect-16.8 P-20ESect-16.9 P-1ESect-16.9 P-2ESect-16.9 P-3ESect-16.9 P-4ESect-16.9 P-5ESect-16.9 P-6ESect-16.9 P-7ESect-16.9 P-8ESect-16.9 P-9ESect-16.9 P-10ESect-16.9 P-11ESect-16.9 P-12ESect-16.9 P-13ESect-16.9 P-14ESect-16.9 P-17ESect-16.9 P-18ESect-16.9 P-19ESect-16.9 P-20ESect-16.9 P-23ESect-16.9 P-24ESect-16.9 P-25ESect-16.9 P-26ESect-16.9 P-27ESect-16.9 P-28ESect-16.9 P-29ESect-16.9 P-30ESect-16.9 P-31ESect-16.9 P-32ECh-16 P-1RCCCh-16 P-2RCCCh-16 P-3RCCCh-16 P-4RCCCh-16 P-5RCCCh-16 P-6RCCCh-16 P-7RCCCh-16 P-8RCCCh-16 P-9RCCCh-16 P-10RCCCh-16 P-11RCCCh-16 P-12RCCCh-16 P-13RCCCh-16 P-14RCCCh-16 P-15RCCCh-16 P-16RCCCh-16 P-1RQCh-16 P-2RQCh-16 P-3RQCh-16 P-4RQCh-16 P-5RQCh-16 P-6RQCh-16 P-7RQCh-16 P-8RQCh-16 P-9RQCh-16 P-10RQCh-16 P-11RQCh-16 P-12RQCh-16 P-13RQCh-16 P-1RECh-16 P-2RECh-16 P-3RECh-16 P-4RECh-16 P-5RECh-16 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