حلول تمارين المحاضرة الاولى للفصل الثاني

تمارين محاضرة ( 1 )
1.If w=cos?(x+y)+sin?(x-y)then show that (?^2 w)/(?x^2 )=(?^2 w)/(?y^2 ).
?w/?x=-sin?(x+y)+cos?(x-y) ? (?^2 w)/(?x^2 )=-cos?(x+y)-sin?(x-y)
?w/?y=-sin?(x+y)-cos?(x-y) ? (?^2 w)/(?y^2 )=-cos?(x+y)-sin?(x-y)

So (?^2 w)/(?x^2 )=(?^2 w)/(?y^2 )

2.Find (?^2 z)/?x?y if z=x^2 sin?(2x-3y).
نشتق الدالة اولا بالنسبة لـ y ونتعامل مع x كثابت
?z/?y=-3x^2 cos?(2x-3y)
نشتق ثانيا بالنسبة لـ x ( مشتقة حاصل ضرب دالتين )
(?^2 z)/?x?y=6x^2 sin?(2x-3y)-6x cos?(2x-3y)

3.Show that f(x,y)=e^3x sin??3y ?,f(x,y)=x^3 -??3xy^2 ? satisfy Laplace^ s equation.
f(x,y)=e^3x sin??3y ?
?f/?x=3e^3x sin?3y ? (?^2 f)/(?x^2 )=9e^3x sin?3y
?f/?y=3e^3x cos?3y ? (?^2 f)/(?y^2 )=-9e^3x sin?3y

(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )=9e^3x sin?3y-9e^3x sin?3y=0
So f(x,y)=e^3x sin??3y ? satisfy Laplace^ s equation.



f(x,y)=x^3 -??3xy^2 ?
?f/?x=3x^2-3y^2 ? (?^2 f)/(?x^2 )=6x
?f/?y=-6xy ? (?^2 f)/(?y^2 )=-6x

(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )=6x-6x=0
So f(x,y)=x^3 -??3xy^2 ? satisfy Laplace^ s equation.
4.If z=ln?(xy) and x=r sin?? , y=r cos?? then show that ?z/??=2 cot?2? .
?z/??=?z/?x×?x/??+?z/?y×?y/??
?z/?x=y/xy=( 1 )/x=( 1 )/(r sin?? )
?z/?y=x/xy=( 1 )/y=( 1 )/(r cos?? )
?x/??=r cos?? and ?y/??=-r sin??
?z/??=( 1 )/(r sin?? )×r cos??+( 1 )/(r cos?? )×(-r sin?? )
?z/??=( cos?? )/sin?? -( sin??)/cos?? =(cos^2??-sin^2??)/sin??? cos?? ? =cos?2?/((1?2) sin?2? )=2 cot?2?