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Chapter 14: Q 28. (page 1154)

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

Short Answer

Expert verified

$鈭� 路 F (x, y, z) = 0$

$鈭� 脳 F (x, y, z) = (- 2 y - 2 z) i + (- 2 z - 2 x) j + (- 2 x - 2 y) k$

Step by step solution

01

Given Information

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

02

Finding the Divergence and Curl

Find the divergence of the given vector function as shown.

$鈭� 路 F (x, y, z) = \frac{鈭�}{鈭� x} F_{1} + \frac{鈭�}{鈭� y} F_{2} + \frac{鈭�}{鈭� z} F_{3}$

Evaluating the divergence

$鈭� 路 F (x, y, z) = 0 + 0 + 0 = 0$

Find the curl of the given vector function as shown.

$鈭� 脳 F (x, y, z) = (\frac{鈭�}{鈭� y} F_{3} - \frac{鈭�}{鈭� z} F_{2}) i + (\frac{鈭�}{鈭� z} F_{1} - \frac{鈭�}{鈭� x} F_{3}) j + (\frac{鈭�}{鈭� x} F_{2} - \frac{鈭�}{鈭� y} F_{1}) k$

Evaluating the curl

$= (- 2 y - 2 z) i + (- 2 z - 2 x) j + (- 2 x - 2 y) k$

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Most popular questions from this chapter

Find the integral of $f (x, y, z) = z^{3} + z (x^{2} + 2^{y})$ on the portion of the unit sphere that lies in the first octant, above the rectangle $[0, \frac{1}{2}] 脳 [0, \frac{1}{3}]$ in the XY-plane.

Compute n for the surface S in Exercise 12.

What is the difference between the graphs of

$G (x, y, z) = 2 i 鈭� 3 j + z k a n d F (x, y, z) = 鈭� 2 i + 3 j 鈭� z k$

$Compute the divergence of the vector fields in Exercises 17 鈥� 22 . F (x, y) = {yx}^{2} i 鈭� x cosy j$

How would you show that a given vector field in ${鈩�}^{2}$ is not conservative?

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