Chapter 14: Q 35 (page 1154)

$\int \int_{S} c u r l F (x, y, z) \cdot n d S$ , where $F (x, y, z) = (5 y, 5 x, z^{2})$
$S$ is the portion of the unit sphere $x^{2} + y^{2} + z^{2} = 1$ above
the plane $z = \frac{1}{2}$ , and $n$ is the upwards-pointing normal vector.

Short Answer

Expert verified

The required the necessary integral is:

$\iint_{S} curl F (x, y, z) \cdot n d S = 0$

Step by step solution

Given information

Consider the vector field below:

$F (x, y, z) = <5 y, 5 x, z^{2}> .$

The goal is to calculate the integral $\iint_{S} curl F (x, y, z) \cdot n d S$ , with the following surface $S$ definition:

The surface $S$ is the region of the unit sphere $x^{2} + y^{2} + z^{2} = 1$ above the plane $z = \frac{1}{2}$ with the normal vector pointing upwards.

Find the vector field's curl first.

$F (x, y, z) = <5 y, 5 x, z^{2}>$

A vector field's curl $F (x, y, z) = F_{1} (x, y, z) i + F_{2} (x, y, z) j + F_{3} (x, y, z) k$ is defined as follows:

After that, the vector field's curl $F (x, y, z) = <5 y, 5 x, z^{2}>$ will be,

$c u r l F (x, y, z) = |\begin{matrix} i & j & k \\ \frac{ϑ}{ϑ x} & \frac{ϑ}{ϑ y} & \frac{ϑ}{ϑ z} \\ 5 y & 5 x & x^{2} \end{matrix}| = [\frac{ϑ (z^{2})}{ϑ y} - \frac{ϑ (5 x)}{ϑ z}] i - [\frac{ϑ (z^{2})}{ϑ x} - \frac{ϑ (5 y)}{ϑ z}] j + [\frac{ϑ (5 x)}{ϑ x} - \frac{ϑ (5 y)}{ϑ y}] k = [0 - 0] i - [0 - 0] j + [5 - 5] k = 0 i - 0 j + 0 k = 0$

Find the necessary integral

The value of $curl F (x, y, z) \cdot n$ will then be,

$c u r l F (x, y, z) \cdot n = 0 \cdot n = 0$

The required integral will then be,

$\int \int_{S} c u r l F (x, y, z) \cdot n d S = \int \int_{S} 0 d S = 0$

As a result, the necessary integral is:

$\iint_{S} curl F (x, y, z) \cdot n d S = 0$

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$\int \int_{S} c u r l F (x, y, z) \cdot n d S$ , where $F (x, y, z) = (5 y, 5 x, z^{2})$
$S$ is the portion of the unit sphere $x^{2} + y^{2} + z^{2} = 1$ above
the plane $z = \frac{1}{2}$ , and $n$ is the upwards-pointing normal vector.

Short Answer

Step by step solution

Given information

Find the vector field's curl first.

Find the necessary integral

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∫∫ScurlF(x,y,z)·ndS, where F(x,y,z)=(5y,5x,z2)Sis the portion of the unit sphere x2+y2+z2=1abovethe plane z=12, and nis the upwards-pointing normal vector.

Short Answer

Step by step solution

Given information

Find the vector field's curl first.

Find the necessary integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Statistics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

$\int \int_{S} c u r l F (x, y, z) \cdot n d S$ , where $F (x, y, z) = (5 y, 5 x, z^{2})$
$S$ is the portion of the unit sphere $x^{2} + y^{2} + z^{2} = 1$ above
the plane $z = \frac{1}{2}$ , and $n$ is the upwards-pointing normal vector.