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MATH1211/LSweek31/TNK/1011(2)
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1211: Multivariable Calculus
Lecture Summary – Week 31
March 28, 2011
Section 6.1
. We recalled
Definition 1.1
(p.364) of the scalar line integral. We noted that
is
the notation for the scalar line integral, but in order to compute it we need to write it as
so that we can integrate w.r.t.
, where
is the parameter of the path
.
We noted that the path of the line integral can be
piecewise
, which is a union of a finite number
of
paths joining together (p.365).
We learned
Definition 1.2
(
vector line integral
) (p.366). We noted the difference between its
notation and that of the scalar line integral: for the scalar line integral, the thing to be integrated is
where
is a scalar valued function and
is the arclength element along the path; for the
vector line integral the thing to be integrated is the dot product
, where
is a vector valued
function, and
is a vector element along the path.
We noted that the vector line integral
has at least three different formulations:
, where everything is written in terms of the parameter
of the path
.
, where
denotes the unit tangent vector along the path
(p.367).
, where
+
. (More generally, if
is a
vector field on
where each
is a scalar function, then
see p.368).
We went through
Example 5
(p.369)
We learned
reparametrization of paths
(
Definition 1.3
, p.369), and that the scalar line integral is
not changed when the path is reparametrized (
Theorem 1.4
, p.370). The same is true for vector line
integrals if the orientation of the path keeps the same as before, but the value of the vector line
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 Spring '11
 wang
 Multivariable Calculus, Scalar, Vector field, vector line, UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS

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