Difference between revisions of "Line integral"
(Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero.) |
(line integral is the area under the function and along the curve. The familiar integral is simply the line-integral using the x-axis as the curve.) |
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| − | A '''line integral''' of a [[vector]] function ''f'' along a line or curve segment ''C'' is the | + | A '''line integral''' of a [[vector]] function ''f'' along a line or curve segment ''C'' is the summation of the values it takes along the curve. It can be expressed in any of these three equivalent ways: |
:<math>\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}</math> | :<math>\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}</math> | ||
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:<math>\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}</math> | :<math>\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}</math> | ||
| − | Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero. | + | Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero. For a conservative field, the difference in potential between the endpoints of a curve equals the line integral along that same curve. |
| − | A '''line integral''' of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the line. | + | A '''line integral''' of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the curve. Put another way, the line integral is the area under the function and along the curve. The familiar, basic integral is simply the line-integral using the x-axis as the curve. |
[[Category:vector analysis]] | [[Category:vector analysis]] | ||
[[Category:calculus]] | [[Category:calculus]] | ||
[[Category:mathematics]] | [[Category:mathematics]] | ||
Revision as of 19:56, December 26, 2009
A line integral of a vector function f along a line or curve segment C is the summation of the values it takes along the curve. It can be expressed in any of these three equivalent ways:
- <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>
Example: The work done on a particle to move it from one point to another is the line integral of the force on the particle along the curve of its motion:
- <math>\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}</math>
Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero. For a conservative field, the difference in potential between the endpoints of a curve equals the line integral along that same curve.
A line integral of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the curve. Put another way, the line integral is the area under the function and along the curve. The familiar, basic integral is simply the line-integral using the x-axis as the curve.