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Q. 15

Expert-verified

Calculus

Found in: Page 1140

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Given an integral of the form $\int_{c} F \cdot d r$ , what considerations would lead you to evaluate the integral with Stokes’ Theorem?

If right-hand side of Stokes' Theorem is used, the result obtained is the same with a single area integral.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

An integral of the form $\int_{C} F \cdot d r$ .

Step 2. Stokes' Theorem

"Let $S$ be an oriented, smooth or piecewise-smooth surface bounded by a curve $C$ . Suppose that $n$ is an oriented unit normal vector of $S$ and $C$ has a parametrization that traverses $C$ in the counterclockwise direction with respect to $n$ . If a vector field $F (x, y, z) = F_{1} (x, y, z) i + F_{2} (x, y, z) j + F_{3} (x, y, z) k$ is defined on $S,$ then, $\int_{c} F (x, y, z) \cdot d r = \int \int_{S} c u r l F (x, y, z) \cdot n d S "$ .

Step 3. Understading the orientation

If $S$ is an oriented, smooth or piecewise-smooth surface bounded by a curve $C$ , Stokes' Theorem relates a line integral of a vector field around the boundary curve $C$ to a surface integral of the curl of the vector field. When the line integral has piecewise-continuous boundary, for example, if the boundary curve $C$ is a rectangle or triangle , it requires the evaluation of several smooth pieces.If you use the right-hand side of Stokes' Theorem, you obtain the same result with a single area integral.

Most popular questions for Math Textbooks

Integrate the given function over the accompanying surface in Exercises 27–34.
$f (x, y, z) = x^{2} - y + 3 z$ , where S is the portion of the plane with equation $2 x - 6 y + 3 z = 1$ whose preimage in the xz plane is the region bounded by the coordinate axes and the lines with equations z = 4 and x = z.

$F = <x \ln (x z), 5 z, \frac{1}{y^{2} + 1}>$ , where S is the region of the plane with equation $12 x - 9 y + 3 z = 10$ , where $2 \leq x \leq 3$ and $5 \leq y \leq 10$ , with n pointing upwards.

Give an example of a field with positive divergence at (1, 0, π).

Compute dS for your parametrization in Exercise 7.

$F (x, y, z) = i + j + k$ , where S is the lower half of the unit sphere, with n pointing outwards.

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