vcprac01 - form. (a) The line passes through the points P...

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The University of Sydney School of Mathematics and Statistics Practice Session 1 MATH2061: Vector Calculus Summer School 2012 1. Evaluate i C φ ds if φ = xy + z and C is the straight line segment from (1 , 1 , 1) to ( 2 , 2 , 5). 2. Describe geometrically the vector Felds determined by each of the following vector functions: (a) F = 3 i + 2 j ; (b) F = x i y j . 3. Evaluate c (2 x y + 4) dx + (5 y + 3 x 6) dy around a triangle in the xy -plane with vertices at (0 , 0) , (3 , 0) , (3 , 2), traversed in the anti-clockwise direction. 4. Evaluate i C F · d r , where F = z i + x j + y k and r ( t ) = t i + t 2 j + t 3 k ; t : 0 1. 5. Calculate the work done by the force Feld F = 3 xy i 2 j in moving from A : (1 , 0 , 0) to B : (2 , 3 , 0) (a) along the straight line AB ; (b) along the straight lines from A to D : (2 , 0 , 0) and then from D to B ; (c) along the piece of the hyperbola x 2 y 2 = 1 , z = 0 from A to B . Revision questions 6. Let u = 2 i 2 j + k , v = i + 3 j k and w = 2 j + 3 k . ±ind ( u · v ) w , u ( v · w ) , ( u × v ) · w , u · ( v × w ) , ( u × v ) × w , u × ( v × w ) . 7. ±ind the equations of the straight lines that satisfy each of the following sets of conditions. Give answers in vector form, parametric scalar form and Cartesian
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Unformatted text preview: form. (a) The line passes through the points P (3 , 3 , 5) and Q (2 , 6 , 1) . (b) The line passes through the point P ( 1 , 2 , 3) and is parallel to the line given by r = (2 i j + k ) + (3 i + j 2 k ) t. (c) The line passes through the point P (1 , 1 , 1) and is perpendicular to the plane 2 x + 3 y z = 4 . Copyright c c 2012 The University of Sydney 1 8. A particle moves along a curve whose parametric equations are x ( t ) = e t , y ( t ) = 2 cos 3 t, z ( t ) = 2 sin 3 t, where t is the time. (a) Determine its velocity vector and acceleration vector. (b) Find the magnitudes of the velocity and acceleration at t = 0 . 2...
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vcprac01 - form. (a) The line passes through the points P...

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