QUESTION 1 Converting from Rectangular Coordinate to Polar Coordinate Choose one . 4 points Change p = (1, v3) from rectangular coordinate to polar coordinate O P = (8, ") =(7:2) O P = (8, T) = (7:2) O P = (8,r) = (7:4) QUESTION 2 Converting from Polar to Rectangular Coordinate Choose one . 4 points Change p = (#, 2) from polar coordinate to rectangular coordinate O P(z, y) = (1, V3) O P(I, y) = (v/3,1) O p(I, y) = (2,7) QUESTION 3 Converting from Cylindrical Coordinate to rectangular Coordinate Choose one . 4 points Change p = (1, #, 2) from cylindrical to rectangular coordinate O V2 v2 P= (-. , 2) O P = (V2 V2,2) O D = 2 ,1)QUESTION 21 Derivative of a scalar function with respect to a vector (Jacobian) Choose one . 4 points If s(x) = x2 + 2 + (2 - 1)2 where a = y , then 2 calculate as O = [2x 2y 2(z-1)] O = 2: 2y 2:] O QUESTION 22 Derivative of a scalar function with respect to a vector (Jacobian) Choose one . 4 points If f(x) = xty+z where = of y , then calculate ax Z O of =[y+z z+z x+y] 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) = ac2 + 2xcy + 22 where x = y , then 2 calculate ag O ag = [2x+ 2y 2x 2=] O ag = [2z+ 2y 2y 2=] O 21 + 2y 2y QUESTION 24 Quadric Forms Choose one . 4 points Express the quadric form f (ac, y) = a2 + 4ay + 592 in matrix form , f(x) = x Ax where where x = y Z and calculate of(a) Ox Or O 21 =[27 + 4y 4z + 10y] O 91 = [27 + By 82 + 10y] OQUESTION 25 Quadric form Choose one . 4 points Express the quadric form f(x, y) = 3x2 + 2xy + 692 in matrix form , f(x) - x Ax where where x = y 2 and calculate = O =[6x + 4y 4x + 12y] O = 26 O of = [fix + 2y 2r + 12y]QUESTION 4 Converting from rectangular Coordinate to Cylindrical Coordinate Choose one . 4 points Change p(1, 1, 1) from rectangular to cylindrical coordinate (p, 4, 2) O p = (1, #, 1) O p = (1, 7, 1) O P =(2 -, QUESTION S Converting from Spherical Coordinate to rectangular Coordinate Choose one . 4 points Change p = (1, 7, 7) from spherical to rectangular coordinates. O P(z, y, z) = (-1,0, 0) O P(z, y, =) = (0, 0, 0) O P(z, y, z) = (1, 0, 0) 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 = (T,0, 0) =(25.7) O D = (7, 0, 0) = (V3 7.7) O P = (7,8,0) = (= =)QUESTION 7 Computing the Gradient off a Scalar field Choose one . 4 points Given the scalar field f(x, y, z) = 22 + 32 + 2x2 +3, calculate grad( f) = Vf = of afit ay of k + az O Vf = 21 + 2y7 + Azk O Vf = 2ri + 2yj + Azk O Vf = 2r +2y + 4z QUESTION 8 Computing the gradient of a Scalar Field Choose one . 4 points Given the scalar field f ( x, y, z) = x+ 2y+ xz+2, calculate grad(f) = Vf = of + dy . + of k az Off = 1+2j+k O Vf = 3+1 O V/ = ( 1+z)i +2j+ zk QUESTION 9 Computing Convectionivalive Choose one . 4 points Let the temperature field be T(x, y, z) = x + 2y+ z , find the convection derivative u . VT = Ur or + Wy dy aT aT + Uz Oz if the velocity field is u = (2, -1, a O I+4 OQUESTION 10 Computing the Curl of a Vector Field Choose one . 4 points Given the vector field u = (ur, Wy , Uz) = x2i + xzj + xy2k calculate curl u = V x u O Vxu= (2ry)i - y]] - zk O Vxu = (2ry - z)i - y j + zk O Vxu = (2ry - z)ity) -zk QUESTION 11 Computing the Curl of a Vector Field Choose one . 4 points Given the vector field u = (ux, Wy, Uz) = xyi + xzj + 2xyk calculate curl u = V x u O Vxu=1 -2 + (2-z)k O Vxi= 3x - 2,j + (2-1)k O Vxu =1 +2 + (2+z)k QUESTION 12 Computing the Divergence of a Vector Field Choose one . 4 points Given the vector field u(ux, Wy, Uz) = x2i + 22 j - xy3k, calculate div u = V.u = auz Our + ay ax az O divi = V - i = 21 O divi = V . i= 2r+ 2z O divi = V-1 =1QUESTION 13 Computing the Divergence of a Vector Field Choose one . 4 points Given the vector field u(Ur, Wy. Uz) = xi + yzj + yzk, calculate div u = V auI ary ouz Or + ay + az O divi = V-1=ityitz O diva= y+ z O diva=itytz QUESTION 14 V Computing the Laplacian of a Scalar Field Choose one . 4 points Given the scalar field f(x, y, z) = a+ + 2+1 compute the Laplacian 2 22 f a2 f a2 f + = + az2 O Vf =4+6y O f = 2 + 6y + 2: O Vf =4+2yQUESTION 15 Computing the Laplacian of a Scalar Field Choose one . 4 points Given the scalar field f (ac, y, z) = xztay2+ azz compute the Laplacian 2 a2 f = 2 2 f 22 f ar2 dy 2 az2 O f = Gry' + 2r O Vf = Gry' + 21 + 21 O V f = 21 +21 QUESTION 16 Computing the Laplacian of a Vector Field Choose one . 4 points Given the vector field u(ur, Wy, Uz) = (3x y, yz, 3xz2) , calculate 2_ the Laplacian of u , V u that is uxhit ( Vug ) i t ( Vuz ) k O Vu= 6 + 6xk O 1 = 6yi + 6zk O Vu= bi + 6kQUESTION 17 Computing the Laplacian of a Vector Field Choose one . 4 points Given the vector field u(ur, Wy, Uz) = (x2y2, y2 z, zz2). calculate 2. the Laplacian of u , Vu that is UI hit ( V uy ) it ( Vuz ) k O Vu= (23 +2y )i + 2zj + 2xk O Vu= (23 +2y) )i + 2z) + 2k O Vu= (217 + 2y) )i + 3] + 2k QUESTION 18 Derivative of a vector with respect to a vector Choose one . 4 points Calculate if w = W1 W2 W1 = + 2y where W2 = 3x + 4y and u = O O Thi OQUESTION 19 Derivative of a vector with respect to a vector Choose one . 4 points Calculate aw W1 , if w = W2 where W1 = y + Z W 2 = + y + 2z and u = Z O - (8 8 8) O = O =QUESTION 20 Derivative of a vector with respect to a vector. Choose one . 4 points W1 Calculate oi , if w = W2 W3 where W1 = 2xy + Z + 22 202 = y 2x + e + x z W3 = ac and u = y Z O 2y 2 2y Oui 22 O 2y 1 1 22 2y I 2y 21 1 1 22