QUESTION 6 Converting from rectangular Coordinate to Spherical Coordinate Choose one . 4 points Change p = (1, 1, 0) from rectangular to spherical coordinates. O P = (r, 0, $) = (2, 7' 7) O p = (r, e, 0) = (v2, 4 O P = (T, 0, $) = (V2 7. 7)QUESTION 7 Computing the Gradient of a Scalar field Choose one . 4 points Given the scalar field f (x, y, z) = a2 + y2 + 2z2 +3, calculate grad( f) = Vf of it ay ask + az O Vf = 21 + 2yj + 4zk O Vf = 2ri + 2yj + 4zk O Vf =2r + 2y + 4zQUESTION 8 Computing the gradient of a Scalar Field Choose one . 4 points Given the scalar field f (x, y, z) = x+ 2y+ az+2, calculate of grad(f) V of of K + Ox 2 + dy + az O Vf = 1+2j +k O Vf = 3+1 O Vf = (1+z )1+2j+IkQUESTION 9 Computing Convection derivative Choose one . 4 points Let the temperature field be T(a, y, z) = x+ 2y + z , find the convection derivative aT aT u . VT = ux Ox + uy dy + uz Oz if the velocity field is u = (2, -1, x O 1+4 O I 4QUESTION 10 V Computing the Curl of a Vector Field Choose one . 4 points Given the vector field u = (Ux, Wy, Uz) = x2i + xzj + xy2k calculate curl u = V x u O V xi = (2ry)i - y]] - zk O V xu = (2ry - x)i - y j + zk O V xi = (2ry - x)ity j - zkQUESTION 11 Computing the Curl of a Vector Field Choose one . 4 points Given the vector field u = (Ux, Wy, Uz) = xyl + xzj + 2xyk calculate curl u = X u O V xu= xi - 2yj + (2 - 1)k O V xi = 3ri - 2yj + (2- x)k O V xi = ri + 2yj + (2+ x)kQUESTION 12 Computing the Divergence of a Vector Field Choose one . 4 points Given the vector field u(ux, Wy, Uz = x2i + 22 j - xy3 k, calculate div u = V . ul ouz dur + ay duy + ax Oz O divi = V . i = 2r O divi = V . i = 2r + 2z O divi = Vi=rQUESTION 13 V Computing the Divergence of a Vector Field Choose one . 4 points Given the vector field u(ux, Wy . Uz) = xi + yzj + yzk, calculate div u = V Our + ay duy + Juz ax az O divi = V. u=ity+ zk O divi=ytz O divi=ltytzQUESTION 14 Computing the Laplacian of a Scalar Field Choose one . 4 points Given the scalar field f ( x, y, z) = a2+ y' + 2+1 compute the Laplacian 2 a2 f a2 f 22 f + 2z2 O f = 4+ 6y O f = 2r + 6y + 2z O if = 4+2yQUESTION 15 Computing the Laplacian of a Scalar Field Choose one . 4 points Given the scalar field f ( x, y, z) = xztay+ xz2 compute the Laplacian 2 a2 f a2 f a2 f + + dy2 + 2z2 O Vf = Gry3 + 213 O Vf = Gry3 + 2x3 + 21 O V f = 2r' + 21QUESTION 16 Computing the Laplacian of a Vector Field Choose one . 4 points Given the vector field u(ux, Wy, Uz = (3x y, yz, 3xz2) , calculate the Laplacian of u , Vu that is V ux)it ( V uy )i t ( Vuz ) k O Vu = Gi+ 6ck O Vu = Gyi + 6rk O Vu = Gyi + 6kQUESTION 17 V Computing the Laplacian of a Vector Field Choose one . 4 points Given the vector field u(ux, Wy, Uz) = (x y2, y2 z, acz2). calculate 2 the Laplacian of u , V u that is 2 2 2 ux )it ( Vug ) j + ( Vuz ) k O Vu= (213 + 2y?)1+ 2zj + 2ck O Vu = (213 + 2y?)i + 2zj + 2k O Vi = (213 + 2y3)2 + =] + 2kQUESTION 18 Derivative of a vector with respect to a vector Choose one . 4 points W1 Calculate if W = W2 W1 = + 2y where W 2 3x + 4y and u = O = (2 4 ) O OQUESTION 19 Derivative of a vector with respect to a vector Choose one . 4 points W1 Calculate , if w W 2 where W1 y + Z 2 2 = y + 2z and u = y Z O = 0 0 0) O O =Choose one . 4 points W1 Calculate du , if w = W2 W3 where 2xy Z W1 = 22 W2 - y 2/2 + W3 = 2x and u = y Z 21 2 21 2y 2z I O 2y I 2z O 2y 21 1 2zQUESTION 21 Derivative of a scalar function with respect to a vector (Jacobian) Choose one . 4 points If s(a) = a2 +2 + ( 2 - 1)2 where x = y then Z calculate O as - = [2x 2y 2(2 -1) ] O =[2x 2y 2=] O 21 2y 2zQUESTION 22 V Derivative of a scalar function with respect to a vector (Jacobian) Choose one . 4 points If f (x) = xtytz where = , then calculate of y Z O of O = O of = 1 1 1]QUESTION 23 Derivative of a scalar function with respect to a vector (Jacobian) Choose one . 4 points If g(ac) = 22 + 2xy + zz where x = y , then ag calculate ax O ag = [2r + 2y 2x 2z] O ag = [2x + 2y 2y 2z] O 2x + 2y 2y 2zQUESTION 24 Quadric Forms Choose one . 4 points Express the quadric form f (x, y) = x2 + 4xy + 5y2 in matrix form , f(a) = x Ax where where a = y Z and calculate of( x) a ( & ] Ax ) Ox ax O af = [2r + 4y 4x + 10y] O 21 = [2x + 8y 8z + 10y] O f(z) = 1AZ = ( ;)#8 =( 8)