Chapter 14: Q 27. (page 1154)

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

Short Answer

Expert verified

$\nabla \cdot F (x, y, z) = 3$

$\nabla \times F (x, y, z) = i + j + k$

Step by step solution

Given Information

The given expression is $F (x, y, z) = (x - y) i + (y - z) j + (z - x) k$

Finding the Divergence and Curl

Find the divergence of the given vector function as shown

$F (x, y, z) = (x - y) i + (y - z) j + (z - x) k$

localid="1651905788821" $\nabla \cdot F (x, y, z) = \frac{\partial}{\partial x} F_{1} + \frac{\partial}{\partial y} F_{2} + \frac{\partial}{\partial z} F_{2}$

Evaluating the divergence

localid="1651905798631" $\nabla \cdot F (x, y, z) = 1 + 1 + 1 = 3$

Find the curl of the given vector function as shown.

localid="1652073592150" $\nabla \times F (x, y, z) = (\frac{\partial}{\partial y} F_{3} - \frac{\partial}{\partial z} F_{2}) i + (\frac{\partial}{\partial z} F_{1} - \frac{\partial}{\partial x} F_{3}) j + (\frac{\partial}{\partial x} F_{2} - \frac{\partial}{\partial y} F_{1}) k = i + j + k$

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

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,z=x-yi+y-zj+z-xk

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