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

$\nabla \cdot F (x, y) = - 2$

$\nabla \times F (x, y) = - 3 k$

Step by step solution

Given Information

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

Finding the Divergence and Curl

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

$\nabla \cdot F (x, y) = \frac{\partial}{\partial x} F_{1} + \frac{\partial}{\partial y} F_{2}$

Evaluating the divergence

$\nabla \cdot F (x, y) = \frac{\partial}{\partial x} (3 x + 4 y) + \frac{\partial}{\partial y} (x - 5 y)$

$\nabla \cdot F (x, y) = 3 - 5 = - 2$

Find the curl of the given vector function as shown.

$\nabla \times F (x, y) = (\frac{\partial}{\partial y} F_{3} - \frac{\partial}{\partial z} F_{2}) i + (\frac{\partial}{\partial z} F_{1} - \frac{\partial}{\partial x} F_{3}) j + (\frac{\partial}{\partial x} G_{2} - \frac{\partial}{\partial y} G_{1}) k$

Evaluating the curl

$\nabla \times F (x, y) = 0 i + 0 j + (1 - 4) k$

$\nabla \times F (x, y) = - 3 k$

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

Short Answer

Step by step solution

Given Information

Finding the Divergence and Curl

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Find the divergence and curl of the following vector fields.Fx,y=3x+4yi+x-5yj

Short Answer

Step by step solution

Given Information

Finding the Divergence and Curl

One App. One Place for Learning.

Most popular questions from this chapter

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Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

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