Hi All,
I want to know the physical significance of Gradient, Curl and Divergence in Vector Calculus.
Can anyone explain me ?

 

Vector calculus is an enormous subject, so you should be reading a textbook for a full treatment.

The whole of vector calculus is about the mathematical description of what are called conservative systems. These systems are the ones normally used in the physical sciences to model physical reality.

So the divergence theorem is a statement of the law of conservation of mass or linear momentum in (fluid) mechanics

The statement curl(f )= 0 is equivalent to saying the flow is laminar or to statement of the law of conservation of angular momentum.

Since in multidimensional work many infinitely tangents can be drawn at a point on a curve, we need a way to distinguish and select a particular one.
This is the function of the gradient, which selects the one pointing in the direction of maximum slope.

Where exactly are you coming from?

 

Hi Studiot,
Thanks for the reply.

Actually I am tying to study "Theory of Electromagnetism". So when I was reading Maxwell's equations in differential form, I was trying to understand the physical significance of those operators ( Gradient, Curl and Divergence) in order to understand it better.
I referred few books but I didn't get physical significance of these operators.
Can you please suggests me any book in which these operators are very well explained along with their physical significance.

Thanking you in advance.

 

Get thee a copy of Feynman, Richard, Lectures on Physics, Vol II.

 

Hi Studiot,
Thanks for the reply.

Actually I am tying to study "Theory of Electromagnetism". So when I was reading Maxwell's equations in differential form, I was trying to understand the physical significance of those operators ( Gradient, Curl and Divergence) in order to understand it better.
I referred few books but I didn't get physical significance of these operators.
Can you please suggests me any book in which these operators are very well explained along with their physical significance.

Thanking you in advance.

Further to Papabravo's recommendation of the Feynman Lectures series (well worth reading if you have a general interest in science) is to get a copy of Erwin Kreyszig's Advanced Engineering Mathematics book. Kreyszig deals with the mathematical constructs from the engineers perspective and allows you to get a "feel" for what the constructs describe. The book is aimed at general engineering, so there it is not just confined to the EE/EM world, but the vector clac section will give you feel for what these constructs describe. Obviously being a maths text, it will also equip you with the skills to work with the vector calc operators.

Dave

 

You should realise that vector calculus methods are very neat and packaged, but when they actually want to put numbers in and get numbers out, engineers use simpler (non vector) formulae. So any good text will describe both approaches.

Div, Grad, Curl and all that.
An Informal text on Vector Calculus

by H M Schey

This little book uses electrical theory as its base example to introduce and develop the subject of Vector Calculus, showing the links between the integral and differential forms of presentation.

Engineering Field Theory

by A J Baden Fuller

This book starts from the engineering/physics viewpoint and introduces and explains pre vector and then vector mathematics as needed for the physical tasks in hand.
It is a gentle introduction which is well explained.

Electromagnetics

J D Krauss

This has a very good introductory chapter on vector mathematics and then proceeds to use them to develop electrical theory in a wide range of situations, maxwells equations, semiconductor physics, waveguides and antenna, radar etc. It is the most modern and contains some computer programs.
Unusually however, Krauss doesn't only use the esoteric vector formulae - it is a very practical book and he also uses simpler appropriate formulae to calculate results.

 

Thanks to all of you for the information.
I will refer the books mentioned by you to get the feel of vectors.
Thanks again.

 

....there's an old book

Barkeley's phisics ...it's quite clear in particlar explaining the Gradient concept

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