Question

Would the sum of two conservative vector fields on a region be conservative on that region too? Please explain.

Answer 1

Hi, @qwerty54321
What does it mean, for a vector field to be conservative?

Answer 2

what field of math is this?

Answer 3

Calculus...?

Answer 4

It looks like @qwerty has left this thread. I'll do my best to explain things... feel free to check for errors...

Answer 5

A vector field \(\large \vec R \) is conservative if there exists a function such that
\[\huge \nabla f = \vec R \]

Answer 6

Now, suppose we have two vector fields \(\large \vec{R_1}\) and \(\large \vec{R_2}\)
Then there exist functions such that
\[\huge \nabla f_1 = \vec{R_1}\]\[\huge \nabla f_2=\vec{R_2}\]

Answer 7

Two vector fields which are *conservative* mind...

Answer 8

Then, taking the sum \[\huge \vec R = \vec{R_1}+\vec{R_2}\]

Answer 9

Consider the function \[\huge f = f_1 + f_2\]
Then clearly
\[\huge \nabla f = \nabla(f_1 + f_2)=\nabla f_1 + \nabla f_2\]

Answer 10

Then \[\huge \nabla f =\vec{R_1}+\vec{R_2}=\vec{R} \]

thus proving that the sum of two conservative vector fields is also conservative, as there is a function whose gradient is that sum.

Answer 11

thank you so much for the explanation; i understand now! :)

Answer 12

No problem.