Understanding Engineering Mathematics

16.4 Higher order derivatives Ifz=f(x,y)is a function ofxandy,then ∂z ∂x is also a function ofxandyand so may itself be partiall ...

Show thatf(x,y)=ln(x^2 +y^2 )satisfies thepartial differential equation: ∂^2 f ∂x^2 + ∂^2 f ∂y^2 = 0 This is called theLaplace ...

z x (x,y) (x+dx,y) (x+dx, y+dy) 0 y dz 2 = dy dz 1 =∂∂xzdx ∂z ∂x Figure 16.6The total derivative. This is exact. However, if we ...

dzis called thetotal differentialofz.Thedx,dy,dzare not actually numerical quantities, but simply symbolise quantities that can ...

We have (treatingdx,dy, etc. as small increments here) dz=αxα−^1 yβdx+βxαyβ−^1 dy so the relative error inzdue to ‘errors’dx,dyi ...

Note also that these error problems can be taken over directly to problems of small increases – e.g. the expansion of a rectangu ...

(iii)h(x,y,z)= √ x^2 +y^2 +z^2 at the points (−1, 2, 2) and (3, 2, 4) (iv)l(x,y,z)=ex 2 y^4 coszat ( 0 ,− 2 , π 3 ) 2.Sketch the ...

science and engineering. An important example is Laplace’s equation ∂^2 f ∂x^2 + ∂^2 f ∂y^2 = 0 which arises in electromagnetic ...

2.(i) 1 (^0) y 1 3 z x (ii) x z − 3 − 1 0 y 1 3 3 (i) 2x,2y (ii) 1 y ,− x y^2 (iii) 3x^2 + 2 xy, x^2 + 4 y^3 (iv) − x (x^2 +y^ ...

(vi) 0, 1 3 (vii) 0,− 5 e−^2 (viii) 0, 19 6 √ 3 (ix) Doe not exist,− 1 5 5.Answers listed in the orderzx,zy,zz (i) 2x,4y,6z (ii) ...

8.Determinedzfor the functions (i) 2xdx− 6 ydy (ii) 6xy^3 dx+ 9 x^2 y^2 dy (iii) 2 x x^2 +y^2 dx+ 2 y x^2 +y^2 dy (iv) −sin(x+y) ...

17 An Appreciation of Transform Methods 17.1 Introduction We have already met situations in which changing the variable in a mat ...

continuous and discontinuous functions (418 ➤ ) partial fractions (60 ➤ ) completing the square (60 ➤ ) differential equations ...

in doing the integral they play no role in the actual integration (other than getting in the way!), which is basically an integr ...

Don’t let anyone tell you all this is easy! There is a great deal of sophisticated mathematics to deal with in this problem, and ...

is called theLaplace transform off.t/. Other notations used areF(s)(lower case goes to upper case when we transform) orL[f(t)], ...

We have already foundL[t] in Problem 17.1, and as a reminder: L[t]= ∫∞ 0 te−stdt= [ te−st −s ]∞ 0 + 1 s ∫∞ 0 e−stdt(by parts)= 1 ...

Table 17.1 f(t) f(s) ̃ =F(s)=L[f(t)] 1 1 s (s > 0 ) t^1 s^2 (s > 0 ) tn(na positive integer) n! sn+^1 (s >^0 ) eat 1 s− ...

Another well known piecewise continuous function is the square wave, shown in Figure 17.2 and defined by f(t)=− 1 − 1 <t< ...

0 1/t Figure 17.4An infinite discontinuity. Note that the value of the integrals involved is unaffected by the absence of the is ...