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Fundamental Theorem for
Line Integrals

In Calculus I we had the Fundamental Theorem of Calculus that told us how to evaluate definite
integrals
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Here
it is
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Also suppose that f is a function
whose gradient vector, ∇f , is continuous on C
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Also, we did not specify the number of variables for the function since it is really immaterial to
the theorem
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Proof
This is a fairly straight forward proof
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Let’s start by just computing the line integral
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∫ ∇f i d r = f ( r ( b ) ) − f ( r ( a ) )

C

Let’s take a quick look at an example of using this theorem
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Solution
First let’s notice that we didn’t specify the path for getting from the first point to the second point
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The theorem above tells us that all we need are the initial and final
points on the curve in order to evaluate this kind of line integral
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Then,

1
r ( a ) = 1, , 2
2

r ( b ) = 2,1, −1

The integral is then,

⎛ 1 ⎞
∫C ∇f i d r = f ( 2,1, −1) − f ⎜⎝1, 2 , 2 ⎟⎠
⎛
⎛π ⎞ ⎛1⎞ ⎞
= cos ( 2π ) + sin π − 2 (1)( −1) − ⎜ cos π + sin ⎜ ⎟ − 1⎜ ⎟ ( 2 ) ⎟
⎝ 2 ⎠ ⎝2⎠ ⎠
⎝
=4
Notice that we also didn’t need the gradient vector to actually do this line integral
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The important idea from this example (and hence about the Fundamental Theorem of
Calculus) is that, for these kinds of line integrals, we didn’t really need to know the path to get
the answer
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In the first section on line integrals (even though we weren’t looking at vector fields) we saw that
often when we change the path we will change the value of the line integral
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Let’s formalize this idea up a little
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The first one we’ve already seen
before, but it’s been a while and it’s important in this section so we’ll give it again
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Definitions
First suppose that F is a continuous vector field in some domain D
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F is a conservative vector field if there is a function f such that F = ∇f
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We first saw this definition in the
first section of this chapter
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is independent of path if

∫ Fid r = ∫ Fi d r

C1

C

for any two paths C1 and C2 in

C2

D with the same initial and final points
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A path C is called closed if its initial and final points are the same point
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4
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A circle is a simple curve while a figure 8
type curve is not simple
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A region D is open if it doesn’t contain any of its boundary points
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A region D is connected if we can connect any two points in the region with a path that
lies completely in D
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A region D is simply-connected if it is connected and it contains no holes
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With these definitions we can now give some nice facts
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∫ ∇f i d r

is independent of path
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The
theorem tells us that in order to evaluate this integral all we need are the initial and final
points of the curve
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2
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C

This fact is also easy enough to prove
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C

Then using the first fact we know that

C

this line integral must be independent of path
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If F is a continuous vector field on an open connected region D and if F i d r is
C

independent of path (for any path in D) then F is a conservative vector field on D
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If

∫ Fid r

C

is independent of path then

∫ F i d r = 0 for every closed path C
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If

∫ F i d r = 0 for every closed path C then ∫ F i d r

C

is independent of path
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Also
notice that 2 & 3 and 4 & 5 are converses of each other

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