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Line Integral Vector function

Line Integral of Vector Function

You can consider the integral of a vector function F along a curve C. The vector function F has the components  and . The path of the parameterized function is .

The parameter t reaches from a to b.

If you use this vector function then the differential d naturally is dr.

This formular expresses that there is a tangent component t of F along the path.

The trajectory of a particle r^’ along a curve (the path C) inside a vector field (Source: Lucas V. Barbosa)

Now the distance gets the name ds.

The line integral over a scalar field f is the area under the curve C along a surface z = f(x,y) which is described by the field (Source: Lucas V. Barbosa)

The line integral is also equivalent to:

  Example

Evaluate the line integral for the function  for . The path shall be a straigt line .

First, write down a parametric function for the path.

t is from 0 to 1.

The new parameters are:

The derivative of x is

The derivative of y is

Put in these parameters into the function F for each dimension x and y.

First dimension x:

The term  is filled in with  and the term  is filled in with

Second dimension y:

The term  is filled in with  

Put in the parameterized variables in an integral.