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Chapter 14: Q. 26 (page 1132)

Compute the curl of the vector fields:
$F (x, y) = - 4 x^{2} y i + 4 x y^{2} j$ .

Short Answer

Expert verified

The curl of the vector field is,

$(4 y^{2} + 4 x^{2}) k$ .

Step by step solution

01

Step 1. Given Information.

The vector field is,

$F (x, y) = - 4 x^{2} y i + 4 x y^{2} j$ .

02

Step 2. Finding the curl of the vector field.

Consider the vector field:

$F (x, y, 0) = - 4 x^{2} y i + 4 x y^{2} j + 0 k$

If $F (x, y, z) = f_{1} (x, y, z) i + f_{2} (x, y, z) j + f_{3} (x, y, z)$ , then

curl $F = 鈭� 脳 F$

role="math" localid="1650456405592" $= |\begin{matrix} i & j & k \\ \frac{鈭�}{鈭� x} & \frac{鈭�}{鈭� y} & \frac{鈭�}{鈭� z} \\ f_{1} (x, y, z) & f_{2} (x, y, z) & f_{3} (x, y, z) \end{matrix}| = |\begin{matrix} i & j & k \\ \frac{鈭�}{鈭� x} & \frac{鈭�}{鈭� y} & \frac{鈭�}{鈭� z} \\ - 4 x^{2} y & 4 x y^{2} & 0 \end{matrix}| = i (\frac{鈭�}{鈭� y} (0) - \frac{鈭�}{鈭� z} (4 x y^{2})) + j (\frac{鈭�}{鈭� z} (- 4 x^{2} y) - \frac{鈭�}{鈭� x} (0)) + k (\frac{鈭�}{鈭� x} (4 x y^{2}) - \frac{鈭�}{鈭� y} (- 4 x^{2} y)) = (0 - 0) i + (0 - 0) j + (4 y^{2} - (- 4 x^{2})) k = (4 y^{2} + 4 x^{2}) k$

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Most popular questions from this chapter

Make a chart of all the new notation, definitions, and theorems in this section, including what each new thing means in terms you already understand.

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.

If S is parametrized by r(u, v), why is $d S = r_{u} 脳 r_{v}$ the correct factor to use to account for distortion of area?

What is the difference between the graphs of

$G (x, y, z) = - i - j - k a n d F (x, y, z) = i + j + k$

How would you show that a given vector field in ${鈩�}^{3}$ is not conservative?

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