▁with ▁a ▁negative ▁sign ▁so ▁that ▁positive ▁work ▁is ▁a ▁reduction ▁in ▁the ▁potential , ▁that ▁is ▁▁ ▁where ▁C ▁is ▁the ▁traject ory ▁taken ▁from ▁A ▁to ▁B . ▁Because ▁the ▁work ▁done ▁is ▁independent ▁of ▁the ▁path ▁taken , ▁then ▁this ▁expression ▁is ▁true ▁for ▁any ▁traject ory , ▁C , ▁from ▁A ▁to ▁B . ▁ ▁The ▁function ▁U ( x ) ▁is ▁called ▁the ▁potential ▁energy ▁associated ▁with ▁the ▁applied ▁force . ▁ ▁Ex amples ▁of ▁forces ▁that ▁have ▁potential ▁energ ies ▁are ▁gravity ▁and ▁spring ▁forces . ▁ ▁Der iv able ▁from ▁a ▁potential ▁In ▁this ▁section ▁the ▁relationship ▁between ▁work ▁and ▁potential ▁energy ▁is ▁presented ▁in ▁more ▁detail . ▁The ▁line ▁integral ▁that ▁defines ▁work ▁along ▁curve ▁C ▁takes ▁a ▁special ▁form ▁if ▁the ▁force ▁F ▁is ▁related ▁to ▁a ▁scalar ▁field ▁ φ ( x ) ▁so ▁that ▁ ▁In ▁this ▁case , ▁work ▁along ▁the ▁curve ▁is ▁given ▁by ▁ ▁which ▁can ▁be ▁evaluated ▁using ▁the ▁gradient ▁theorem ▁to ▁obtain ▁ ▁This ▁shows ▁that ▁when ▁forces ▁are ▁deriv able ▁from ▁a ▁scalar ▁field , ▁the ▁work ▁of ▁those ▁forces ▁along ▁a ▁curve ▁C ▁is ▁computed ▁by ▁evalu ating ▁the ▁scalar ▁field ▁at ▁the ▁start ▁point ▁A ▁and ▁the ▁end ▁point ▁B ▁of ▁the ▁curve . ▁ ▁This ▁means ▁the ▁work ▁integral ▁does ▁not ▁depend ▁on ▁the ▁path ▁between ▁A ▁and ▁B ▁and ▁is ▁said ▁to ▁be ▁independent ▁of ▁the ▁path . ▁ ▁Pot ential ▁energy ▁U =- φ ( x ) ▁is ▁tradition ally ▁defined