
#1
Dec2412, 08:46 AM

P: 64

I am having difficulties to understand why in mathematics when calculating line integrals using gradient theorem we use F=grad(U), and in physics it is always F=grad(U)? It seems important to me, because I may end up getting answer with opposite sign.
Is it somehow related to Newton's third law? 



#2
Dec2412, 10:03 AM

Sci Advisor
HW Helper
PF Gold
P: 12,016

Only those forces we call "conservative" will be gradients of scalar fields.
You'll meet other types of forces that are not conservative, for example forces of friction. 



#3
Dec2412, 10:07 AM

P: 350

If we put them into a common context, then F is a force field. A line integral of F represents the work done by the force field on a free particle that traverses that path. The amount of work done is equal to the particle's gain in kinetic energy, which is equal to its loss of potential energy. So when physicists flip the sign, they are accounting for changes in potential energy rather than changes in kinetic energy.
Outside this physical context, there is no reason to flip the sign. You just use the fundamental theorem of line integrals. 



#4
Dec2512, 02:01 AM

P: 64

Gradient theorem, why F=grad(U) ? 



#5
Dec2612, 04:20 AM

P: 428

Because forces point downhill. In math, the purer idea is uphill.



Register to reply 
Related Discussions  
Theorem proofs in applied math grad programs  Academic Guidance  8  
closed curve line integral of gradient using Green's Theorem  Calculus & Beyond Homework  2  
Gradient and mean value theorem  Calculus & Beyond Homework  6  
Gradient theorem question  Calculus & Beyond Homework  1  
surface integral, grad, and stokes theorem  Calculus & Beyond Homework  5 