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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

$鈭� 路 F (x, y, z) = 3$

$鈭� 脳 F (x, y, z) = i + j + k$

Step by step solution

01

Given Information

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

02

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

Evaluating the divergence

localid="1651905798631" $鈭� 路 F (x, y, z) = 1 + 1 + 1 = 3$

Find the curl of the given vector function as shown.

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

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Most popular questions from this chapter

Find the work done by the vector field

$F (x, y) = x^{3} y^{2} i + (y - x) j$

in moving an object around the triangle with vertices $(1, 1), (2, 2)$ , and $(3, 1)$ , starting and ending at $(2, 2)$ .

$F (x, y, z) = i + j + k$ , where S is the lower half of the unit sphere, with n pointing outwards.

If curl $F (x, y, z) 路 n$ is constantly equal to $1$ on a smooth surface $S$ with a smooth boundary curve $C$ , then Stokes鈥� Theorem can reduce the integral for the surface area to a line integral. State this integral.

Why is the orientation of S important to the statement of

Stokes鈥� Theorem? What will change if the orientation is

reversed?

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

See all solutions

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