Q. 9 If聽F(x,y,z)=鈭噁(x,y,z)is a con... [FREE SOLUTION]

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Chapter 14: Q. 9 (page 1150)

If $F (x, y, z) = 鈭� f (x, y, z)$ is a conservative vector field, what will div $F$ be?

Short Answer

Expert verified

The value of $d i v F = f_{x x} + f_{y y} + f_{z z}$

Step by step solution

Introduction

Given is a conservative vector field $F (x, y, z) = 鈭� f (X, y, z)$ . The objective is to find $d i v F$ .

Step 2

the divergence of a vector field is $F (x, y, z) = F_{1} (x, y, z) i + F_{2} (x, y, z) j + F_{3} (x, y, z) k$ is defined as follows

$d i v F = 鈭� . F$

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

:

Then, the divergence of vector field $F (x, y, z) = 鈭� f (x, y, z)$ will be

$d i v F = d i v 鈭� f = d i v (\frac{鈭� f}{鈭� x} i + \frac{鈭� f}{鈭� y} j + \frac{鈭� f}{鈭� z} k) = (\frac{鈭�}{鈭� x} i + \frac{鈭�}{鈭� y} j + \frac{鈭�}{鈭� z} k) . (\frac{鈭� f}{鈭� x} i + \frac{鈭� f}{鈭� y} j + \frac{鈭� f}{鈭� z} k) = \frac{鈭�}{鈭� x} (\frac{鈭� f}{鈭� x}) + \frac{鈭�}{鈭� y} (\frac{鈭� f}{鈭� y}) + \frac{鈭�}{鈭� z} (\frac{鈭� f}{鈭� z}) = \frac{{鈭�}^{2} f}{鈭� x^{2}} + \frac{{鈭�}^{2} f}{鈭� y^{2}} + \frac{{鈭�}^{2} f}{鈭� z^{2}} = f_{x x} + f_{y y} + f_{z z}$

Therefore , if $F (x, y, z)$ is a consecutive vector field , then

$d i v F = f_{x x} + f_{y y} + f_{z z}$

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91影视

If $F (x, y, z) = 鈭� f (x, y, z)$ is a conservative vector field, what will div $F$ be?

Short Answer

Step by step solution

Introduction

Step 2

:

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91影视

IfF(x,y,z)=鈭�f(x,y,z)is a conservative vector field, what will div Fbe?

Short Answer

Step by step solution

Introduction

Step 2

:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Calculus

Applied Mathematics

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

If $F (x, y, z) = 鈭� f (x, y, z)$ is a conservative vector field, what will div $F$ be?