If I want to find a function V(r) such that F=−grad V, where F=f(r)hat(r), how do I integrate F along the radial direction?

spockmonkeyqj

spockmonkeyqj

Open question

2022-08-19

If I want to find a function V ( r ) such that F = V, where F = f ( r ) r ^ , how do I integrate F along the radial direction?

Answer & Explanation

ellynnauwh

ellynnauwh

Beginner2022-08-20Added 7 answers

Let for x R d let r = | x | . Note that if g : [ 0 , ) R , then the gradient of the function V ( x ) = g ( | x | ) is given by
V = r r g .
Thus to solve your problem we need to find g such that
r r g ( r ) = f ( r ) r ,
which leads to
g ( r ) = 0 r f ( t ) t d t + C .

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