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Theorems Concerning Vector Fields


Theorem

Given the vector field E → x , y , z = f x , y , z i → + g x , y , z j → + h x , y , z k → . If E → = − ∇ φ , then:
  1. E → is a conservative field.
  2. ∇ φ is the gradient field of E → .
  3. φ is the potential of E → .

The Fundamental Theorem of Line Integrals

Given a vector field E → x , y = f x , y i → + g x , y j → ,where f and g are continuous functions on plane D, which contains the points x 0 , y 0 and x 1 , y 1 . If E → x , y = − ∇ φ x , y for every point in D, then for each smooth curve C in D, which start at x 0 , y 0 and ends at x 1 , y 1 , the following is true: ∫ c E → x , y · d r → = φ x 1 , y 1 − φ x 0 , y 0

Equivalance Theorem

Given a vector field, E → x , y = f x , y i → + g x , y j → , in which f and g are continous functions on a plane D, then the following three statements are equivalent (either they are all true or non of them is true):
  1. E → is a conservative field in D.
  2. ∮ C E → · d r → = 0 for each closed curve C in D.
  3. ∮ C E → · d r → does not depend on the path, for each curve C in D.