Therefore, by the Fundamental Theorem of Calculus,
□
We know that if
F is a conservative vector field, there are potential functions
such that
Therefore
In other words, just as with the Fundamental Theorem of Calculus, computing the line integral
where
F is conservative, is a two-step process: (1) find a potential function (“antiderivative”)
for
F and (2) compute the value of
at the endpoints of
C and calculate their difference
Keep in mind, however, there is one major difference between the Fundamental Theorem of Calculus and the Fundamental Theorem for Line Integrals.
A function of one variable that is continuous must have an antiderivative. However, a vector field, even if it is continuous, does not need to have a potential function.
Applying the fundamental theorem
Calculate integral
where
and
C is a curve with parameterization
without using the Fundamental Theorem of Line Integrals and
using the Fundamental Theorem of Line Integrals.
First, let’s calculate the integral without the Fundamental Theorem for Line Integrals and instead use
[link] :
Integral
requires integration by parts. Let
and
Then
and
Therefore,
Thus,
Given that
is a potential function for
F , let’s use the Fundamental Theorem for Line Integrals to calculate the integral. Note that
This calculation is much more straightforward than the calculation we did in (a). As long as we have a potential function, calculating a line integral using the Fundamental Theorem for Line Integrals is much easier than calculating without the theorem.
[link] illustrates a nice feature of the Fundamental Theorem of Line Integrals: it allows us to calculate more easily many vector line integrals. As long as we have a potential function, calculating the line integral is only a matter of evaluating the potential function at the endpoints and subtracting.
Given that
is a potential function for
calculate integral
where
C is the lower half of the unit circle oriented counterclockwise.
The Fundamental Theorem for Line Integrals has two important consequences. The first consequence is that if
F is conservative and
C is a closed curve, then the circulation of
F along
C is zero—that is,
To see why this is true, let
be a potential function for
F . Since
C is a closed curve, the terminal point
r (b) of
C is the same as the initial point
r (a) of
C —that is,
Therefore, by the Fundamental Theorem for Line Integrals,
Questions & Answers
Biology is a branch of Natural science which deals/About living Organism.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
