Chapter 14: Q. 9 (page 1150)

If $F (x, y, z) = \nabla f (x, y, z)$ 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 $F (x, y, z) = \nabla f (X, y, z)$ . 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

$d i v F = \nabla . F$

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

:

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

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

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

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

Geometry

Pure Maths

Decision Maths

Logic and Functions

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

IfF(x,y,z)=∇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

Geometry

Pure Maths

Decision Maths

Logic and Functions

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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