Chapter 16.8, Problem 2E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use Stokes’ Theorem to evaluate ∫∫* _{s}* curl

**2. F**(*x*, *y, z*) *= x*^{2}sin *z* **i** + *y*^{2} **j** + *xy* **k**. *S* is the part of the paraboloid *z =* 1 - *x*^{2} *- y*^{2} that lies above the *xy*-plane, oriented upward

To determine

**To evaluate**: The expression

Explanation

**Given data:**

Consider the expression for the vector field

And *S* is the part of the paraboloid that is,

Equation (1) lies above the *xy*-plane, oriented upward.

**Formula Used:**

Consider the expression for the Stokesâ€™ theorem,

The paraboloid in equation (2) intersects the *xy*-plane in the circle
*C* must be oriented in the counter-clockwise direction. Therefore, the vector equation of *C* is,

Differentiate equation (4) with respect to *t*,

Find the expression for

Substitute
*x*,
*y* and 0 for *z* in equation (1) to find

Write the expression for the Stokesâ€™ theorem in equation (3),

Substitute

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