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Chapter 14: Q. 23 (page 1150)

In Exercises $21 - 24$ , compute the divergence of the given vector field.
$F (x, y, z) = x e^{x y z} i + y e^{x y z} j + z e^{x y z} k$

Short Answer

Expert verified

The divergence of the given vector field is $3 e^{x y z} + 3 x y z e^{x y z}$

Step by step solution

01

Step 1

Think about the vector field,

$F (x, y, z) = x e^{x y z} i + y e^{x y z} j + z e^{x y z} k$

Our goal is to determine the vector field's divergence.

Make use of the formula, $div F = \frac{鈭� F_{1}}{鈭� x} + \frac{鈭� F_{2}}{鈭� y} + \frac{鈭� F_{3}}{鈭� z}$

In comparison to the supplied vector, $F (x, y, z) = x e^{x y z} i + y e^{x y z} j + z e^{x y z} k$ using the vector $F (x, y, z) = F_{1} (x, y, z) + F_{2} (x, y, z) + F_{3} (x, y, z)$

So, $F_{1} (x, y, z) = x e^{x y z}$

$F_{2} = (x, y, z) = y e^{x y z}$

$F_{3} = (x, y, z) = z e^{x y z}$

02

Calculation

Estimate $div F$

$div F = \frac{鈭� F_{1}}{鈭� x} + \frac{鈭� F_{2}}{鈭� y} + \frac{鈭� F_{3}}{鈭� z}$

$= \frac{鈭�}{鈭� x} (x e^{x y z}) + \frac{鈭�}{鈭� y} (y e^{x y z}) + \frac{鈭�}{鈭� z} (z x^{x y z})$

$= (e^{x y z} + x y z e^{x y z}) + (e^{x y z} + x y z e^{x y z}) + (e^{x y z} + x y z e^{x y z})$

$= 3 e^{x y z} + 3 x y z e^{x y z}$

Therefore,role="math" localid="1650802546816" $div F = 3 e^{x y z} + 3 x y z e^{x y z}$

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Most popular questions from this chapter

Work as an integral of force and distance: Find the work done in moving an object along the x-axis from the origin to $x = \frac{蟺}{2}$ if the force acting on the object at a given value of x is role="math" localid="1650297715748" $F (x) = x \sin x$ .

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
$f (x, y, z) = x y z^{2}$ , where S is the portion of the cone $\sqrt{3} z = \sqrt{x^{2} + y^{2}}$ that lies within the sphere of radius 4 and centered at the origin.

Integrate the given function over the accompanying surface in Exercises 27鈥�34.

$f (x, z) = e^{- (x^{2} + z^{2})}$ , where S is the unit disk centered at the point (0, 2, 0)and in the plane y = 2.

Suppose that an electric field is given by

$E = 2 y i + 2 x y j + y z k$

Compute the flux ${鈭�}_{S} 鈥� E 鈰� n d A$ of the field through the unit cube $[0, 1] 脳 [0, 1] 脳 [0, 1]$ .

Find the work done by the vector field

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

in moving an object around the unit circle, starting and ending at $(1, 0)$ .

See all solutions

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