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Integration of a vector field along a curve
Given a curve in space with starting point A and end point B in parametric form
This curve can be a vector
assign whose peak the curve when varying the parameter from after passes through.
The curve is differentiable if , and differentiable functions of are, and then has the tangent vector
The curve piece is tangent to the curve at that point .
a vector field. We define the line integral of the vector field along the curve as
i.e., is the integral of the scalar product of with . Here is the scalar product
Calculation of work
As an example, consider the work involved in moving a body in a force field is performed. If the body moves a distance , will be a job
performed. You move the body along a curve from point A to point B and adds up the individual distances to get all the work done
If there is a conservative force field, it can be described as a gradient of a scalar field grasp
The work done is then
i.e., with a conservative force field, the work done is path-independent. In particular, the work for a closed curve is zero.
Examples of conservative force fields are the electric field and the gravitational field. A non-conservative vector field is the magnetic field.