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

Step 1. Given Information.

The vector field is,

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

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 $F = \nabla \times F$

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

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the curl of the vector field.

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Compute the curl of the vector fields:Fx,y=-4x2yi+4xy2j.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the curl of the vector field.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

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