Difference between revisions of "Line integral"
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A '''line integral''' is the area under a function along a specified curve. An ordinary or Riemann [[integral]] is a line integral using the x-axis as the line and bound by a given interval on the x-axis. | A '''line integral''' is the area under a function along a specified curve. An ordinary or Riemann [[integral]] is a line integral using the x-axis as the line and bound by a given interval on the x-axis. | ||
| − | + | A '''line integral''' is typically evaluated for the values taken by a three-dimensional vector field along a line or curve segment ''C''. In this common use the line integral of a vector function ''f'' along ''C'' is the following (all three formulations are equivalent); | |
| + | |||
| + | :<math>\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}</math> | ||
| + | |||
| + | :<math>\int_\mathbf{C} f_1s_1 dx + f_2s_2 dy + f_3s_3 dz</math> | ||
:<math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math> | :<math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math> | ||
| + | [[Category:vector analysis]] | ||
[[Category:calculus]] | [[Category:calculus]] | ||
[[Category:mathematics]] | [[Category:mathematics]] | ||
Revision as of 17:47, December 26, 2009
A line integral is the area under a function along a specified curve. An ordinary or Riemann integral is a line integral using the x-axis as the line and bound by a given interval on the x-axis.
A line integral is typically evaluated for the values taken by a three-dimensional vector field along a line or curve segment C. In this common use the line integral of a vector function f along C is the following (all three formulations are equivalent);
- <math>\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}</math>
- <math>\int_\mathbf{C} f_1s_1 dx + f_2s_2 dy + f_3s_3 dz</math>
- <math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math>