Find the flux of the vector field F(x,y,z)=zi+yj+xk over the unit sphere x^2 + y^2 + z^2 = 1?
86765
Plus infinity (It appears!)\n\nWe must compute the Surface Integral of the vector field F overthe surface S of the Sphere.\n\nBy divergence theorem of Gauss it is equal to the \ntriple integral of div F d(x,y,z).\n\nIntroducing spherical coordinates\nx=rCos[u]Sin[v]\ny=rSin[u]Sin[v]\nz=rCos[v]\n\nSubstition of F=(z,y,x) in the formula for divF in spherical coordinates \nand computing the triple integral inside the sphere\n\nIntegral ( -Pi/2 < v < Pi/2 )\n  Integral ( 0 < u < 2Pi )\n    Integral ( 0 < r < 1 )\n      (1/r)(2Cos[v]+cos[u]+2Cos[u]Cos[v])\n    dr\n  du\ndv\n\nThe integral (0<r<1) of (1/r)dr is infinity!\nJust ignore the other parts.
829160
y'=2 cos(7x)*(-sin 7x)*7 By the Chain Rule
292019
If the unit vector has components "a" and "b" (the unknowns for which you want to solve) then your first equation is\n\n1.  a^2 + b^2 = 1\n\nIf you have f and the directional derivative then you can take the partial derivatives of f with respect to x and y (call these f_x and f_y) and get your second equation\n\n2.  a f_x + b f_y = directional derivative\n\nWith two equations in the two unknowns a and b you then try to solve for a and b.
601694
Just solve for h:\nV = πr^2h \n\nDivide both sides by  πr^2\nV / πr^2 = h\n\nSwapping sides:\nh = V / πr^2\n\nSo if V = 121 and r = 3\nh = 121 / 9π\nh = 4.27949958 cm\n\nSorry, but I can't graph the function for you.  It would be a 3D graph with 3 variables (V, h, r).  For example for V = 0, h would be 0 for all values.  Similarly for r = 0, h would be 0 for all values.  Other than that, plug in coordinates for V and r and then graph h on the vertical (z) dimension.\n\nFor the value of h, however it is approx. 4.28 cm.
653586
#1\n\nThe normal to the plane 3x - y + z = 1 is <3, -1, 1> (the coefficients of the variables).\n\nThe line with the same direction as this vector through the point (8, 4, 0) can be defined by the parametric equations:\n\nx = 3t + 8\ny = -t + 4\nz = t\n\n(The coeffiecients for t are that of the vector and the constants are from the point.)\n\nNow to find where the line intersects the coordinate planes, set x, then y, then z as 0.\n\nyz-plane: x = 0\n0 = 3t + 8\nt = -8/3\nSo y = 8/3 + 4 = 20/3\nz = -8/3\n**The point is (0, 20/3, -8/3)\n\nxz-plane: y = 0\n0 = -t + 4\nt = 4\nSo x = 12 + 8 = 20\nz = 4\n**The point is (20, 0, 8)\n\nxy-plane: z = 0\n**This is the given point (8, 4, 0)\n\n\n#2\n\nThe first line contains the vector <1, 6, 2> and the point (2, 2, 2).  The second contains the vector <3, 21, 8> and the point (2, 3, -4).\n\nThe vector between the two points is <2-2, 2-3, 2--4> = <0, -1, 6>.  You can project this vector onto one of the lines.\n\nRemember that the length of the projection of vector b on vector a is equal to the dot product of a and b over the magnitude of a.\n\nIn this case vector b is between the lines and is <0, -1, 6> and vector a is one of the lines; I'll pick line 1 and vector <1, 6, 2>.\n\na dot b = 0*1 + -1*6 + 6*2 = -6 + 12 = 6\n\nmegnitude of a = sqrt(1 + 36 + 4) = sqrt(41)\n\nTherefore, the magnitude of the projection is 6/sqrt(41)\n\nNow you have two sides of a right triangle.  The hypotenuse is so called vector b, so we need its magnitude.\n\nmagnitude of b = sqrt(0 + 1 + 36) = sqrt(37)\n\nNow just find the length of the final side - d, the distance between the skew lines!\n\nd^2 + 36/41 = 37\n\nThen just solve for d and you have your distance.\n\nI hope this helps!  Good luck with your final!\n\nP.S. I double-checked my math, but check it just is case I made a mistake and didn't catch it.
1044611
F = Gm1m2/ r^2\nF = ma  \na = v/t\nv = d/t\na =d/t^2 \nF = m d/t^2  = Gm1m2/ d^2\nm  = (m1 + m2)\nF = (m1 + m2) d/t^2 = Gm1m2 /d^2\nd/m2t^2 + d/m1t^2 = G/d^2\nd/m2 + d/m1 = Gt^2/d^2\nSQRT{ (1/G) d^2 (d/m2 +d/m1)} = t
971390
{3XY^2M/1}-{2M/[X^-1Y^-1]^2\n+{[4X^2Y^2]/M^-1}-\n(3X^(3Y)^(3M)^-1/(XYM)^-2
694104
conjugate(sin(z)) = sin(conjugate(z))
140559
Using the abc method… \n[a]FexOy+[b]H2SO4 -->[c]Fe2(SO4)3+[d]SO2 +[e]H2O\nFe-->ax=2c------eq.1\nO-->ya+4b=12c+2d+e----eq.2\nH-->2b=2e----------eq.3\nS-->b=3c+d----------eq.4\n\nLet a=1\n\nFrom eq.1\nax=2c\nx/2=2/2c\nx/2=c-----eq.5\n\nForm eq.3\n2b=2e\n2/2b=2/2e\ne=b-----eq.6 \n\nFrom eq.2\nYa+4b=12c+2d+e\n1y+4b=12(x/2) +2d+b (Form eq.5&6)\nY+3b=6x+2d\n2d=y-6x+3b\nd=(y+3b-6x)/2------------eq.7\n\nFrom eq.4\nb=3c+d (but c=x/2)\nb=3(x/2) +d\nReplacing eq.4 by eq.7 we will get\nb=3x/2+(y+3b-6x) ½\nb= (3x+y+3b-6x) ½\n2[b=(y+3b-3x) ½]\n2b=y+3b-3x\n3x-y=b-------eq.8\n\nReplacing eq.6 by eq.8\ne=b\ne=3x-y\n\nReplacing eq.7 by eq.8\nd=(y+3b-6x)/2\nd=[y+3(3x-y)-6x]/2\nd=[y+9x-3y-6x]/2\nd= (3x-2y)/2\n\nFrom the above calculation we know that:\na=1, b=3x-y, c=x/2, d=(3x-2y)/2 & e=3x-y. \n\n[a]FexOy+[b]H2SO4 -->[c]Fe2(SO4)3+[d]SO2 +[e]H2O\nWhen replacing the values we’ll get:\n[1]FexOy+[3x-y]H2SO4 -->[x/2]Fe2(SO4)3+[(3x-2y)/2]SO2 +[3x-y]H2O
187220
W=F*d,  d=W/F,  d=600J/400N = 1.5m
606263
1/r^2\n\nIf you ask why, think of intensity as a flux given by \n\nFlux =  Total flow / (cross sectional area)\n\nNow the wave diverges spherically, but the total flow remains constant, owing to conservation laws. \n\nSo flux = Total flow / (Surface area of the sphere)\n\nNote that the total area of the sphere = 4*pi*r^2. \n\nThat gives you the r^2 term in the denominator. \n\nThis is nothing but a demonstration of the inverse squared law, which is much a part of classical theories of gravitation (Newton's law), electrostatics(Coulomb's law) and of the theory of sound intensity.
195576
Y = U + VX^2\nVX^2 = Y - U\nX^2 = (Y-U)/V\nX = sqrt ( (Y-U)/V )
806655
dx=2xlny-2z\ndy=x^2*(1/y)-2z\ndz= x^2lny-2
694807
Use conservation of impulse and conservation of kinetic energy.\n\nmv + 0 = mu + mw\n(1/2)mv2 = (1/2)mu2 + (1/2)mw2\n\nu + w = 1 (consider it as a straight line in a graph)\nu2 + w2 = 1 (consider it as a circle in that graph)\nWhen you solve this set of two equations, you get\nu = 0 and w = 1 (m/s)\n(Find the useful intersection. The other one is is the initial situation.)\n\nTh
585433
The Jacobian determinant for the suggested transformation is:\n\n|d(u,v,w)/d(x,y,z)| = |y x 0 , z 0 x , 0 z y| = -2xyz\n==>\ndudvdw = dxdydz(-2xyz)\n==>\ndxdydz = -dudvdw/(2xyz) = -dudvdw/(2*sqrt(uvw))\n\n(The sign just depends on the order of variables and is not significant.) All variables are positive. The volume is the integral:\n\n&#92;int_1^9 du &#92;int_1^4 dv &#92;int_1^16 dw 1/(2*sqrt(uvw)\n=\n(1/2)\n* &#92;int_1^9 du/sqrt(u) \n* &#92;int_1^4 dv/sqrt(v)\n* &#92;int_1^16 dw/sqrt(w)\n=(1/2) * 2^3 * (3-1)*(2-1)*(4-1) = 24
329292
V=(4/3)*pi*r^3\n\nwhere r is the radius of the sphere and V is volume.
1102035
THE CORRECT ANS IS\n\n                 (x-a)^2+(y-b)^2+(z-c)^2=r^2\n         \n                 where (a, b, c) is the centre of the sphere\n                 and 'r' is the radius \n                  this gives only surface of the sphere\n                  for its interior part '=' is replaced by '<'