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Theorems Concerning Vector Fields
Theorem
Given the vector field
. If
, then:
-
is a conservative field.
-
is the gradient field of
.
-
is the potential of
.
The Fundamental Theorem of Line Integrals
Given a vector field
,where f and g are continuous functions on plane D, which contains the points
and
.
If
for every point in D, then for each smooth curve C in D, which start at
and ends at
, the following is true:
.
Equivalance Theorem
Given a vector field,
, 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):
-
is a conservative field in D.
-
for each closed curve C in D.
-
does not depend on the path, for each curve C in D.