Chemistry

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

x=x(t),y=y(t),z=z(t),tA.ttB..

This curve can be a vector

r(t)=x(t)e1+y(t)e2+z(t)e3

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

F.(x,y,z)=F.1(x,y,z)e1+F.2(x,y,z)e2+F.3(x,y,z)e3

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

I.=C.F.dr=tA.tB.F.(x(t),y(t),z(t))r'(t)dt,

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

F.(x(t),y(t),z(t))r'(t)=F.1x'(t)+F.2y'(t)+F.3z'(t).

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

dW.=F.dr

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

W.=C.F.dr.

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

F.=-φ=-φxe1+φye2+φze3.

The work done is then

W.=-C.φxdx+φydy+φzdz=φ(xA.,yA.,zA.)-φ(xB.,yB.,zB.),

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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