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line_integrals

# line_integrals - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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V11. Line Integrals in Space 1. Curves in space. In order to generalize to three-space our earlier work with line integrals in the plane, we begin by recalling the relevant facts about parametrized space curves. In 3-space, 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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