Complete and incomplete differential

Complete and incomplete differential

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Integration of a vector field along a curve

Given a curve C. 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 t from tA. after tB. passes through.

The curve is differentiable if x(t), y(t) and z(t) differentiable functions of t are, and then has the tangent vector

dr(t)dt=r'(t)=x'(t)e1+y'(t)e2+z'(t)e3 .

The curve piece dr=r'(t)dt is tangent to the curve at that point r(t).

Well be


a vector field. We define the line integral of the vector field F. along the curve C. as


i.e., I. is the integral of the scalar product of F. with dr. Here is the scalar product


Calculation of work

As an example, consider the work involved in moving a body in a force field F.(x,y,z) is performed. If the body moves a distance dr, will be a job


performed. You move the body along a curve C. from point A to point B and adds up the individual distances drto get all the work done


If there is a conservative force field, it can be described as a gradient of a scalar field φ(x,y,z) 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.

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