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

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

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

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

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

Calculation

Estimate $div F$

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

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

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

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

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

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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

Step by step solution

Step 1

Calculation

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In Exercises21-24, compute the divergence of the given vector field.F(x,y,z)=xexyzi+yexyzj+zexyzk

Short Answer

Step by step solution

Step 1

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Logic and Functions

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

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