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Assignment on Numerical Methods

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Added on  2020-04-21

Assignment on Numerical Methods

   Added on 2020-04-21

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Running Head: NUMERICAL METHODS1Numerical MethodsNameInstitution
Assignment on Numerical Methods_1
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 of 13(2,1,2)From above the partial derivatives arefx (x,y,z)= y3z2 - 2x therefore fx (1, -2, 1) = (-2)3(1)2 – (2×1)fx (1, -2, 1) = -10fy (x,y,z)=3xy2z2 + 4zy therefore fy (1, -2, 1) = (3)(1)(-2)2(1)2 + (4×1×-2) =4fy (1, -2, 1)=4fz (x,y,z)= 2xy3z +2y2therefore fz (1, -2, 1)= (2)(1)(-2)3(1) + (2)(-2)2fy (1, -2, 1)= -8The direction derivative is Du f (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)div v = .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
Assignment on Numerical Methods_2
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=ddxxr ̅3+ddyyr ̅3+ddzzr ̅3ddxxr ̅3=xddxr ̅3+r ̅³ddxx=-3xr-4drdx+r-3drdx=ddx(x2+y2+z2)ˆ12therefore ddx=¿3xr-4.xr-1+ r-3=3x²r5+1r3ddyyr ̅3=yddyr ̅3+r ̅³ddyy=-3yr-4drdy+r-3drdy=ddy(x2+y2+z2)ˆ12therefore ddy=¿3yr-4.yr-1+ r-3=3y²r5+1r3ddzzr ̅3=zddzr ̅3+r ̅³ddzz=-3zr-4drdz+r-3drdz=ddz(x2+y2+z2)ˆ12therefore ddz=¿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^kddxddyddzxr ̅³yr ̅³zr ̅³
Assignment on Numerical Methods_3

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