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Chapter 14: Q 26. (page 1154)

Find the divergence and curl of the following vector fields.
$F (x, y) = x^{2} y i + x y^{3} j$

Short Answer

Expert verified

$鈭� 路 F (x, y) = 2 x y + 3 x y^{2}$

$鈭� 脳 F (x, y) = (y^{3} - x^{2}) k$

Step by step solution

01

Given Information

The given expression is $F (x, y) = x^{2} y i + x y^{3} j$

02

Step 2. Finding the Divergence and Curl

Find the divergence of the given vector function as shown.

F (x, y) = x^{2} y i + x y^{3} j

localid="1651905655379" role="math" $鈭� 路 F (x, y) = \frac{鈭�}{鈭� x} F_{1} + \frac{鈭�}{鈭� y} F_{2}$

Evaluating the divergence

localid="1651905645792" role="math" $鈭� 路 F (x, y) = 2 x y + 3 x y^{2}$

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

localid="1651525649420" $鈭� 脳 F (x, y) = 0 i + 0 j + (y^{3} - x^{2}) k$

$鈭� 脳 F (x, y) = (y^{3} - x^{2}) k$

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Most popular questions from this chapter

What are the inputs of a vector field in ${鈩�}^{3}$ ?

Compute dS for your parametrization in Exercise 7.

The current through a certain passage of the San Juan Islands in Washington State is given by

$F = <F_{1} (x, y), F_{2} (x, y)> = <0, 1.152 - 0.8 x^{2}> .$

Consider a disk R of radius 1 mile and centered on this region. Denote the boundary of the disk by localid="1650455155947" $鈭� R$ .

(a) Compute localid="1650455166518" ${鈭�}_{R} (\frac{鈭� F_{2}}{鈭� x} - \frac{鈭� F_{1}}{鈭� y}) d A$ .
(b) Show that ${鈭�}_{鈭� R} F 路 n d s = {鈭�}_{0}^{2 蟺} (1.152 - 0.8 \cos^{2} 胃) \sin 胃 d 胃 .$ Conclude that Green's Theorem is valid for the current in this area of the San Juan Islands.
(c) What do the integrals from Green's Theorem tell us about this region of the San Juan Islands?

$Compute the divergence of the vector fields in Exercises 17 鈥� 22 . F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$

$F (x, y, z) = 鈭� y i + x j 鈭� e^{y z} k$ , where S is the cylinder with equation $x^{2} + y^{2} = 9$ from $z = 2, 4$ , with n pointing outwards.

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