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Chapter 14: Q 25. (page 1154)

Find the divergence and curl of the following vector fields.
$F (x, y) = (3 x + 4 y) i + (x - 5 y) j$

Short Answer

Expert verified

$鈭� 路 F (x, y) = - 2$

$鈭� 脳 F (x, y) = - 3 k$

Step by step solution

01

Given Information

The given expression is $F (x, y) = (3 x + 4 y) i + (x - 5 y) j$

02

Finding the Divergence and Curl

Find the divergence of the given vector function as shown. $F (x, y) = (3 x + 4 y) i + (x - 5 y) j$

$鈭� 路 F (x, y) = \frac{鈭�}{鈭� x} F_{1} + \frac{鈭�}{鈭� y} F_{2}$

Evaluating the divergence

$鈭� 路 F (x, y) = \frac{鈭�}{鈭� x} (3 x + 4 y) + \frac{鈭�}{鈭� y} (x - 5 y)$

$鈭� 路 F (x, y) = 3 - 5 = - 2$

Find the curl of the given vector function as shown.

$鈭� 脳 F (x, y) = (\frac{鈭�}{鈭� y} F_{3} - \frac{鈭�}{鈭� z} F_{2}) i + (\frac{鈭�}{鈭� z} F_{1} - \frac{鈭�}{鈭� x} F_{3}) j + (\frac{鈭�}{鈭� x} G_{2} - \frac{鈭�}{鈭� y} G_{1}) k$

Evaluating the curl

$鈭� 脳 F (x, y) = 0 i + 0 j + (1 - 4) k$

$鈭� 脳 F (x, y) = - 3 k$

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Most popular questions from this chapter

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
$f (x, y, z) = x^{2} - y + 3 z$ , where Sis 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.

Problem Zero: Read the section and make your own summary of the material.

$F (x, y, z) = 鈭� x z i 鈭� y z j + z^{2} k$ , where S is the cone with equation $z = \sqrt{x^{2} + y^{2}}$ between $z = 2, 4$ , with n pointing outwards.

Find the integral of $f (x, y, z) = x - y - z$ on the portion of the plane with the equation

$10 x 鈭� \sqrt{33} y + 36 z = 30$ with 2 鈮� x 鈮� 7 and 1 鈮� z 鈮� 2.

Find the masses of the lamina:

The lamina occupies the region of the hyperboloid with equation $z^{2} + 4 = x^{2} + y^{2}$ that lies above and/or below the disk of radius 5 about the origin in the XY-plane, and the density function, 蟻(x, y,z), is proportional to the distance from the origin.

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