Mathematical Methods for Physics and Engineering - Matematica.NET

1.3 COORDINATE GEOMETRYwith K and φ as given above. Notice that the inverse sine yields two values in the range 0to 2π and that there is no real solution to the original equation if |k| > |K| =(a 2 +b 2 ) 1/2 . ◭1.3 Coordinate geometryWe have already mentioned the standard form for a straight-line graph, namelyy = mx + c, (1.35)representing a linear relationship between the independent variable x and thedependent variable y. Theslopem is equal to the tangent of the angle the linemakes with the x-axis whilst c is the intercept on the y-axis.An alternative form for the equation of a straight line isto which (1.35) is clearly connected byax + by + k =0, (1.36)m = − a band c = − k b .This form treats x and y on a more symmetrical basis, the intercepts on the twoaxes being −k/a and −k/b respectively.A power relationship between two variables, i.e. one of the form y = Ax n ,canalso be cast into straight-line form by taking the logarithms of both sides. Whilstit is normal in mathematical work to use natural logarithms (to base e, writtenln x), for practical investigations logarithms to base 10 are often employed. Ineither case the form is the same, but it needs to be remembered which has beenused when recovering the value of A from fitted data. In the mathematical (basee) form, the power relationship becomesln y = n ln x +lnA. (1.37)Now the slope gives the power n, whilst the intercept on the ln y axis is ln A,which yields A, either by exponentiation or by taking antilogarithms.The other standard coordinate forms of two-dimensional curves that studentsshould know and recognise are those concerned with the conic sections – so calledbecause they can all be obtained by taking suitable sections across a (double)cone. Because the conic sections can take many different orientations and scalingstheir general form is complex,Ax 2 + By 2 + Cxy + Dx + Ey + F =0, (1.38)but each can be represented by one of four generic forms, an ellipse, a parabola, ahyperbola or, the degenerate form, a pair of straight lines. If they are reduced totheir standard representations, in which axes of symmetry are made to coincide15