Vector Field Line Integral Formula

This integral usually called a riemann integral can be thought of as an integral over a path in mathbbr1 since an interval or collection of intervals is really the only kind of path in mathbbr1.

Vector field line integral formula. In single variable calculus you learned how to integrate a real valued function f x over an interval ab in mathbbr1. The value of the line integral is the sum of values of the field at all points on the curve weighted by some scalar function on the curve commonly arc length or for a vector field the scalar product of the vector field with a differential vector in the curve. So a line integral is sum of arclength multiplied by the value at that point. Where x xt yt.

Want to know more on the line integrals in vector calculus. Where ci are partitions from a to b spaced by dsi. If fpxyzqxyzrxyz then in the figure above it is shown that c is traversed in the counter clockwise direction. Line integral formula for vector field for a vector field with function f.

Vectors a vector is a ray that starts at a point x y z and goes in the direction xi yj zk. The line integral cf tds exists. Notice that hence from previous work on line integrals we have line integrals of vector fields extend to three dimensions. We draw vector field with evenly spread points for visual purposes but you should imagine the field as a continuum.

Such an integral cftds is called the line integral of the vector field f along the curve c and is denoted as c p dxqdyrdz. Where x xt yt zt. A vector field is the compilation of these vectors at every point. The line integral of a scalar field 3 superb examples.

U r n r n a line integral along with a smooth curve c u in the direction r is defined as. We can also write line integrals of vector fields as a line integral with respect to arc length as follows c f dr c f t ds c f d r c f t d s where t t t t is the unit tangent vector and is given by. Line integral of a vector field. Note that the arc length can also be determined using the vector components st xti ytj ztk.

Here is a visual representation of a line integral over a scalar field.