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18.02 Multivariable Calculus
Fall 2007
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Line Integrals in Space
1.
Curves
in
space.
In order to generalize to threespace our earlier work with line integrals in the plane, we
begin by recalling the relevant facts about parametrized space curves.
In 3space, a vector function of one variable is given as
It is called continuous or differentiable or continuously differentiable if respec
tively x(t), y(t), and z(t) all have the corresponding property. By placing the
vector so that its tail is at the origin, its head moves along a curve C as t varies.
This curve can be described therefore either by its position vector function (I),
or by the three parametric equations
The curves we will deal with will be finite, connected, and piecewise smooth; this means
that they have finite length, they consist of one piece, and they can be subdivided into a
finite number of smaller pieces, each of which is given as the position vector of a continuously
differentiable function (i.e., one whose derivative is continuous).
In addition, the curves will be oriented, or directed, meaning that an
arrow has been placed on them to indicate which direction is considered to
be the positive one. The curve is called closed if a point
P moving on it
always in the positive direction ultimately returns to its starting position,
as in the accompanying picture.
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 Integrals, Multivariable Calculus

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