LINEARALGEBRA AND CALCULUS
PARITIAL DIFFERENTIATION
Tutorlal-4
2
1. If U =log(x +y' +z-3xyz), showthat
2. If U=y' where x= log y=e find
ar
ây
du
d
0Uty+:)
0z
3. If U= log(x+ y+z ) where x = e' y= sin l z= cosl
4. If x+3x'y +6xy²+y' =1 find
5.
Find
du
dx
-9
find
du
dt
dy
dx
if u= xlog(xy) where x' +y' +3xy =1
6. If u = f(2x-3y, 3y -4z, 4z- 2x) then show that
1du + 1 du 1du
2 dx 3 dy 4 dz
=0
u, v, w
7. If u=x-2y, v=x+y+Z, w=x-2y +3zthenfind J(, y, z)
8. If x=rcos , y=rsin then show that
=1
9. If x= ul-v)y=uw then prove that JJ' =1
10. Find J|
where u =x'-y',v=2xy and x=r cos 6, y=rsin 0
11. Show that the functions u=
1-xy
V=tan1x+ tan yare functionally dependent. Find the relation between
them.
12. Expand e*Sin y in powers of xand y.
13. Expand f,y)= x*y+3y - 2in powers of (x-1) and (y+2).
14. Find Maxima and minima of function x+3ry'-3x -3y² +4.
15. The temperatureT at any point (x, y, z) in space isT= 400xyz. Find the highest temperature on the surface of
the unit sphere x+y +z' =1
16. Find the minimum value of x +y' +z",given that xy +yz +zx =3a*
17. Find the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid
=1.