Question

How can we find conservative zero in maths class 9th Question

Answer 1

ANSWER. ..........
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IN DETAIL
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Why is the curl of a conservative vector field zero?

There has been a couple of answers that state that by definition a conservative field is that which can be written as the gradient of a scalar function or directly that whose curl is zero. Although this are all correct, I don´t think they are very clarifying. I´ll try to explain in a physical intuitive way.

Let´s define a conservative vector field as one whose linear integral between any two points does not depend in the path chosen, which is also a correct definition.
Now it´s a bit more clear why it is called conservative field. From this it is easy to deduce that the line integral of a circular path must be zero in a conservative field.

Let´s see now Stoke´s Theorem :


Stoke´s Theorem states that the surface integral of the curl of a vector field over a surface equals the line integral over the border of this surface .

If the surface integral of the curl must be zero for any surface chosen, it must be zero everywhere.

But I wanted to give a clarifying answer, so I will explain why the Stoke´s Theorem is true (I´m not sure this qualifies as demonstration so let´s leave it in explanation).

For any given loop:
If we divide in two loops:


This holds for any number of divisions:


Now for a sufficiently small loop:


And if the area is small enough we have:







So with this and the Riemann integral we have the Stoke´s Theorem.
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