Line Integrals

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Introduction:

Let C be a cure in space. The orientation of the curve C is defined by a direction along C. Therefore are two possible directions along C namely A to B and B to A. If the direction from A to B is defined as the positive direction. The Parametric representation of curve r(t) = x(t) i + y(t) j + z(t) k. A region R in which every closed curve can be contracted to a point without passing out of the region is called simply connected region; otherwise it is called a multiply connected region. For example the region interior to a circle or a sphere is a simply connected region.

Explanation of line integral:

Any integral which is to be evaluated along a curve is called a line integral . Let F(t) = F1i + F2 j + F3 k be a vector point function defined along a curve C. Let r = x i + y j + z k be the position vector of any point on this curve. Let the arc length along this curve be measured from a fixed point A. If s denotes the arc length from A to any point P(x, y, z) we know that `(dr)/(ds)` = t is a unit vector. along the tangent to the curve at P. The component of F along the tangent given ...
... by F `(dr)/(ds)`. The integral of this component along C measured from the point A to the point B is given by `int_A^B F` `(dr)/(ds)` ds. This integral is called the line integral of F along C. This integral is also called the tangential line integral of F along C.

Scalar function:

The scalar function of line integral is `int_c( F. (dr)/(ds))ds = int_c F. dr`

Note 1: if F = F1 i + F2 j + F3 k

r = x i + y j + z k

So `int_c F.dr` = `int_c ` (F1dx + F2dy + F3 dz)

Note 2: if the equation of the curve is given in parametric form say x = x(t), y = y(t) and z = z(t) and the parametric values at A and B are t = t1 and t = t2 then

` int_c F. dr = int_(t_1)^(t_2)(F_1(dx)/(dt) + F_2 (dy)/(dt) + F_3 (dz)/(dt)) dt`

Application of line integral:

F is a force acting upon a particle which moves along a curve C in space and r be the position vector of the particle at a point on C. Then work done by the particle at C is F.dr and the total work done by F in the displacement along a curve C is given by the line integral `int_c F.dr`

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