082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

240 Chapter 7 Vectors
2 Calculate the distance travelled bythe robot.
3 Now consider the problem ofvisitingallofthe
positionsA, B,C,Dand Eonetime only, with the
robot travelling the least distance. The robot must
enter through the doorway and finishat the
assembly line. Bytrialand errorproduce a solution
to the problem, usingthe computer to plot each
trial. Expressyourfinal calculated solution in
vector notation.
7.4 SCALARFIELDSANDVECTORFIELDS
Imaginealargeroomfilledwithair.Atanypoint,P,wecanmeasurethetemperature, φ,
say. The temperature will depend upon whereabouts in the room we take the measurement.
Perhaps, close to a radiator the temperature will be higher than near to an open
window. Clearly the temperature φ isafunctionofthe position ofthe point.Ifwelabel
the point by its Cartesian coordinates (x,y,z), then φ will be a function of x,y and z,
that is
φ = φ(x,y,z)
Additionally, φ may be a function of time but for now we will leave this additional
complication aside. Since temperature is a scalar what we have done is define a scalar
ateach pointP(x,y,z) inaregion. This isanexample of ascalar field.
Alternatively,supposeweconsiderthemotionofalargebodyoffluid.Ateachpoint,
fluid will be moving with a certain speed in a certain direction; that is, each small fluid
elementhasaparticularvelocity,v,dependinguponwhereaboutsinthefluiditis.Since
velocity is a vector, what we have done is define a vector at each pointP(x,y,z). We
now have a vector function ofx,y andz, known as avector field. Let us write
v = (v x
,v y
,v z
)
so that v x
,v y
and v z
arethei,j and kcomponents respectively ofv, thatis
v=v x
i+v y
j+v z
k
We note that v x
, v y
and v z
will each bescalar functions ofx,y andz.
Engineeringapplication7.5
ElectricfieldstrengthEandelectricdisplacementD
Electrostaticsisthestudyoftheforceswhichstationarypositiveandnegativeelectric
charges exert upon one another. Consider Figure 7.24 which shows a single positive
charge placed at O. The presence of this charge gives rise to an electric force field
around it. Faraday introduced the idea of lines of force to help visualize the field.
At any point, P, there exists a vector which gives the direction and magnitude of
the electrostatic force at P. Because all the lines of force emerge radially from O,
the direction of the electrostatic force is radially outwards. It can be shown that the
magnitudeoftheforceisinverselyproportionaltothesquareofthedistancefromO.