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1. Surface integral (a) [10 marks] Evaluate the surface integral //s y2 dS. S is the part of the plane . + y + 2 = 1 that lies in the first octant. (b) [10 marks] Evaluate the surface integral F - ds for the given vector field F = - S + zk. S is the part of the plane . + y + 2 = 1 that lies in the first octant. 2. Green's theorem (a) [10 marks] Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ey dar + 2.re³ dy C is the square with sides x = 0, . = 1. y = 0 and y = 1. (b) [10 marks] Use Green's Theorem to find the work done by the force F(r.y) = in moving a particle from the origin along the to (1.0), then along the line segment to (0,1), then back to the origin along the y-axis. 3. Curl and divergence (a) [10 marks) Find the curl and the divergence of the vector fields F(x.y.z) =rydi+ny?:j+ = and F(x.y.z) = e singi + cos yj + zk. (b) [10 marks] Determine whether or not the vector field F = e2i + it .re²k is conservative. If it is, find a function f such that F = Vf. 4. Vector field integrals (a) [10 marks] Use Stokes' Theorem to evaluate //s curlF - dS where F(x.y,2) = + sin ry j + ryck. and S is the part of the cone y2 = >2 + 22 that lies between the planes y I 0 and y = 3. oriented in the direction of the positive y-axis. (b) [10 marks] Use the divergence theorem to calculate the surface integral F - dS, that S is, calculate the flux of F = xyi - . !] j - x2 yzk across S. S is the surface of the solid bounded by the hyperboloid 22 + y - z2 - 1 and the planes 2 = 2 and 2 - 2. 5. Laplace transform (a) [10 marks] Find the Laplace transform L(sintcost + (2t). (b) [10 marks] Find the inverse Laplace transform + 2s s2 + 2 - s2 + 4s - 4

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Differential Equations Problems
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