Vector Line Integrals

vector line integrals:

In integral calculus, we have defined indefinite integral, definite integral, line integrals, surface integral and volume integral for a real valued function. If `vecf` (t) and `vecF` (t) be two vector valued functions, such that `d/dt` `vecF` (t) = `vecf` (t) then `vecF` (t) is called the integral of `vecf` (t) with respect to t and this is denoted in symbol by `int vecf` (t)dt = `vecF` (t). In general if `d/dt[vecF(t) + vecc] = vecf(t) ` where `vec c` is  an orbitrary vector independent of t, then `int vecf(t)dt = vecF(t) + vec c`

Here, `vecF` (t) is called indefinite integral of `vecf` (t) and `vecc` is an orbitrary constant of integration.  The definite integral for vector valued function is `int_a^bvecf(t)dt`  = `[ vecF(t)+vecc]^b_a`  = `vec F` (b) -` vecF` (a)

Explanation of Vector Line Integral:

Any integral which is to be evaluated along a curve is called a line integral . Let `vecF (t) = F_1veci + F_2 vecj + F_3 veck ` be a vector point function defined along a curve C.  Let `vecr = xvec i + y vecj + z veck` 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 `(dvecr)/(ds)`  = vect is a unit vector. along the tangent to the curve at P. The component of `vecF` along the tangent given by `vecF` `(dvecr)/(ds)`. The integral of this component along C measured from the point A to the point B is given by  `int_A^B vecF`  `(dvecr)/(ds)` ds. This integral is called the line integral of `vecF` along C. This integral is also called the tangential line integral of `vecF` along C.

Between, if you have problem on these topics how to solve complex rational expressions, please browse expert math related websites for more help on formula for conditional probability.

Scalar function:

The scalar function of line integral  is `int_c( vecF. (dvecr)/(ds))ds = int_c vecF. dvecr`

Note 1: if `vecF (t) = F_1veci + F_2 vecj + F_3 veck `

`vecr = xvec i + y vecj + z veck`

`dvecr = dx veci+dy vecj+dzvec k`

`vecF.dvecr = F_1dx+F_2dy+F_3dz`

So `int_c vecF.dvecr` = `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 vecF. dvecr = int_(t_1)^(t_2)(F_1(dx)/(dt) + F_2 (dy)/(dt) + F_3 (dz)/(dt)) dt`

Application of Vector 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`