Solution :
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To use change of variable theorem,
14) Consider the parallelepiped D determined by the vectors (2,-1,2), (1,3, 1), and (2,-1,1). Let...
5s , y s-t to compute the (1 pt) In this problem we use the change of variables x integral(xy) dA, where R is the parallelogram formed by (0, 0), (5,1), (8, -2), and (3, -3) д(х, у) д(s,t) First find the magnitude of the Jacobian, ,b = Then, with a = and d с 3 b JRC+y) dA = ) dt ds =
q4 please thanks (1) Let A - (0,0), B- (1,1) and consider the veetor field f(r, y,z)vi+aj. Evaluate the line integral J f.dr )along the parabola y from A to B and (i)along the straight line from A to B. Is the vector field f conservative? (2) For the vector feld f # 22(r1+ gd) + (x2 + y2)k use the definition of line integral to (3) You are given that the vector field f in Q2 is conservative. Find...
(1) Let u = (-1,2) and v = (3, 1). (a) (5] Find graphically the vector w = (2u - v). (b) (5] Find algebraically the vector z=3u - 2 (2) (a) [5] Write u ='(1, -5, -1) as a linear combination of v1 = (1,2,0), v2 = (0,1,-1), V3 = (2,1,1). (b) (5] Are the 4 vectors u, V1, V2, V3 linearly independent? Explain your answer. (C) (5) Are the 2 vectors V, V3 linearly independent? Explain your answer....
(1 point) This problem will illustrate the divergence theorem by computing the outward flux of the vector field F(x, y, z) - 2ri + 5y + 3-k across the boundary of the right rectangular prism: -3 <<6, -15y<3,-425 oriented outwards using a surface integral and a triple integral over the solid bounded by rectangular prism. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the prism to be...
(1,3), с %3D (2,1), d (3,4) (1,2), b (4,2), f (5,3) and (5,5). Let 5. Let a = е 3 - {a, b, c, d, e, f, g} be the set of these 7 points. We define the following partial order on S: We have (r, y)(', y) iff x< x and y < / Draw the Hasse diagram of S S 6. We consider the same partial order as in Problem 5, but it is now defined on R2....
3. Let X.. U(-1,1) for i € {1, 2, 3). (a) Write down the expected value and variance of X,. Sketch the pdf fx. [3] (b) Let Y = X1 + X2. Compute the pdf fy of Y and sketch it. Using fy, or otherwise, compute the expected value and variance of Y. (7) (c) Let Z = X1 + X2 + X3 = Y + X3. Using exactly the same technique that you used in part (b), it can...
1. Are £i and C2 skew lines? Explain your answer and find the distance between them if they are skew lines. 3 marks 2. Let S be the region given by S-((z, y) E R: z2 + y2 4,z? + y2-4y2 0,#2 0, y 20} 1 mark (a) Sketch the region S; (b) Consider the change of variables given by u2 , a2 +y-4y. Describe the region S as set in terms of the variables u and v. Call this...
(1 point) This problem will illustrate the divergence theorem by computing the outward flux of the vector field F x, y, z = 2ī + 4j + k across the boundary of the right rectangular prism: 1 sx <5,-2 Sys3,-33z37 oriented outwards using a surface integral and a triple integral over the solid bounded by rectangular prism. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the prism...
n - meraymowa:)--00 [1] [ Let the vectors x, y and z be x = -2 y=1tz= -1 [3] [2] Find r. s and t such that y + z = x O (r, s, t) = (-2, -1, 1) O (r, s, t) = (-2, 1, 1) O (r, s, t) = (-2, 1,-1) (r, s, t) = (2, 1,-1) m Consider the set S = {w,x,y,z} of vectors in R3, S = { 121, Let V = span...
please solve all with detailed steps. thank you! Find the mass, and the center of mass of the solid cone D with density p(x, y, z) = 1 bounded by the surface z = 4- x2 + y2 and z = 0 1) 2) Evaluate dA where R is the square with vertices (0,0), (1,–1), (2,0), and (1,1) x+y+1 (Hint: use a convenient change of variables) 3) Evaluate the line integral (x - y+ 2z)ds where C is the circle...