I want to know about the functions which is path independent and path dependent. Please describe me about this.

# Calculus

### #1

Posted 16 November 2015 - 02:24 PM

### #2

Posted 17 November 2015 - 08:45 AM

What is this in relation to?

Are you asking about conservative and non conservative fields or forces?

Gravity is conservative, friction is non conservative.

Or are you asking about zero v nonzero curl?

### #3

Posted 19 November 2015 - 07:12 PM

Yes I want to know about conservative and non conservative fields. Also I want to know about zero and nonzero curl.

### #4

Posted 19 November 2015 - 09:01 PM

So how did your query arise?

You need to put more in here to get more out.

### #5

Posted 20 November 2015 - 05:05 PM

My previous question posted by mistake. I considered that there is a relation between them. Actually I just wanted to know about the line integral. Is every line integral independent of path?

**Edited by SF Shawn, 20 November 2015 - 05:13 PM.**

### #6

Posted 21 April 2016 - 12:47 PM

It is linked to the line only. But the the "integrand" itself may be parametric. Line is the path.

### #7

Posted 22 September 2016 - 08:08 PM

My previous question posted by mistake. I considered that there is a relation between them. Actually I just wanted to know about the line integral. Is every line integral independent of path?

I am puzzled by this question. Any text that introduces path integrals will tell you the **conditions** under which this is true. If a path integral, is independent of the path, then we could **define** a function F(x,y,z) by "F(x,y,z) is the integral from (0, 0, 0) to (x, y, z)". But then it would follow that , , . But then we would have to have , , and . (A differential for which that is true is called an **exact** differential.)

As an easy example, if we integrate on the **straight line** from (0, 0, 0), we can take as parametric equations x= y= z= t so the integral becomes .

But if we integrate that same integrand from (0, 0, 0) to (1, 1, 1) by taking the path (0, 0, 0) to (1, 0, 0), then to (1, 1, 0), then to (1, 1, 1) we get:

On (0, 0, 0) to (1, 0, 0) take x= t, y= 0, z= 0. The integral becomes .

On (1, 0, 1) to (1, 1, 0) take x= 1, y= t. z= 0. The integral becomes .

On (1, 1, 0) to (1, 1, 1) take x= 1, y= 1. z= t. The integral becomes .

The integral from (0, 0, 0) to (1, 1, 1), along that path is the sum 0+ 1+ 2= 3.

Here, of course,

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