Assignment on Numerical Methods

Added on - 21 Apr 2020

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Running Head: NUMERICAL METHODS1Numerical MethodsNameInstitution
NUMERICAL METHODS2Numerical MethodsQuestion 1a.Evaluate the directional derivative of scalar functionf(x,y,z)=xy3z2+ 2zy2- x2at the point (1, -2, 1) in the direction of13(2,1,2)From above the partial derivatives arefx(x,y,z)= y3z2- 2xthereforefx(1, -2, 1) = (-2)3(1)2– (2×1)fx(1, -2, 1) = -10fy(x,y,z)=3xy2z2+ 4zythereforefy(1, -2, 1) = (3)(1)(-2)2(1)2+ (4×1×-2) =4fy(1, -2, 1)=4fz(x,y,z)= 2xy3z +2y2thereforefz(1, -2, 1)= (2)(1)(-2)3(1) + (2)(-2)2fy(1, -2, 1)= -8The direction derivative isDuf(1, -2, 1) =-10(23¿+ 4(13¿– 8(23¿=323b.Find an expression for the divergence of the vector fieldv(x,y,z) = (x3y2-z, yz, x2z)at the point (1, 3, -1)divv=.vTaking partial derivative of the above equationfx(x3y2-z) + fy(yz) + fz(x2z)=3x2y2+z +2xat the point (1, 3, -1),.v=(3)(1)2(3)2-1 +2 = 28c.F(x,y,z)=(2xy2–yx, 2yx2+2yz2-xz, 2zy2-xy)find×F and׿×F)×F=ijkxyz2xy2yx2yx2+2yz2xz2zy2xy
NUMERICAL METHODS3= (4zy-x-4yz-x)i-(-y-o)j+(4yx-z-4xy-x)k=-2x^i+y^j-(z+x)^k׿×F) = 0F(x,y,z)=r/r³Calculate curl;r= (x2+y2+z2)ˆ12r=(x,y,z)thus F=(xr³,yr³,zr³¿.F=ddxxr̅3+ddyyr̅3+ddzzr̅3ddxxr̅3=xddxr̅3+r̅³ddxx=-3xr-4drdx+r-3drdx=ddx(x2+y2+z2)ˆ12thereforeddx=¿3xr-4.xr-1+ r-3=3x²r5+1r3ddyyr̅3=yddyr̅3+r̅³ddyy=-3yr-4drdy+r-3drdy=ddy(x2+y2+z2)ˆ12thereforeddy=¿3yr-4.yr-1+ r-3=3y²r5+1r3ddzzr̅3=zddzr̅3+r̅³ddzz=-3zr-4drdz+r-3drdz=ddz(x2+y2+z2)ˆ12thereforeddz=¿3zr-4.zr-1+ r-3=3z²r5+1r3.F=(3x²r5+1r3¿+(3y²r5+1r3¿+(3z²r5+1r3¿CITATION Ric12 \l 1033(Hamming, 2012)Divergence.v=^i^j^kddxddyddzxr̅³yr̅³zr̅³
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