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Unformatted text preview: August 15, 2009 Mathematics of Physics and Engineering II: Homework problems Homework 1. Problem 1. Consider four points in R 3 : P (1 , 1 , 1) , Q (- 1 , , 2) , R (1 ,- 1 ,- 1) , S (1 + a, , 1- 2 a ), where a is a real number. (1) Compute the coordinates of- PQ ,- PR ,- PS (2) Compute the value of a so that the angle RPS is the right angle. (3) Compute the area of the triangle PQR . (4) Compute the equation of the plane containing points P,Q,R . (5) Compute the volume of the parallelepiped spanned by the vectors- PQ,- PR,- PS (it will be a function of a ). (6) For what value of a will the point S lie in the same plane as the points P,Q,R ? (7) Write the vector parametric equation of the line that passes through the point (1 , , 1) and is perpendicular to the plane through points P,Q,R , and compute the coordinates of the point of intersection of the line and the plane. Problem 2 . A particle moves so that its position at time t is given by the vector function r ( t ) = h 1- t 2 ,t 3 , 1 + t 2 i , t . Compute: (1) Coordinates of the particle at time t = 1: (2) Velocity of the particle for t 0: (3) Speed of the particle for t 0: (4) Acceleration of the particle for t 0: (5) Vector parametric equation of the tangent line to the trajectory at (0 , 1 , 2). (6) The coordinates of the point of intersection of the trajectory with the plane z- x = 2. (7) Distance traveled (arc length) from t = 0 to t = 1. Homework 2. Solve each problem and write 1-2 variations on a problem of your choice. Problem 1. Consider a function f ( x,y ) = 2 x 2- xy + y 2- x + y- 1 . (1) Compute the gradient of f . (2) Compute the rate of change of f at (1 , 1) in the direction toward the origin. Is the function increasing or decreasing in that direction? (3) Determine the direction of most rapid decrease of f at (1 , 1) and compute the rate of change of the function in that direction. (4) (Warning: this one can be time-consuming) Write the equation of the path of steepest ascent on the surface z = f ( x,y ) starting from point (0 , , 1). What path will you get on the topographic map? Problem 2. Evaluate the following integrals: (1) Z C (2 y 2 + 2 xz ) dx + 4 xydy + x 2 dz, where C is the path x ( t ) = cos t,y ( t ) = sin t,z ( t ) = t, t 2 . (2) I C y 2 dx + x 2 dy , where C is the boundary of the rectangle with vertices (1 , 0) , (3 , 0) , (3 , 2) , (1 , 2), oriented counterclockwise. (3) I C ydx- zdy + ydz, where C is the ellipse x 2 + y 2 = 1;3 x + 4 y + z = 12 oriented counterclockwise as seen from the point (0 , , 1000). Problem 3. Compute the following quantities using a suitable integral: 1 (1) The mass of the curve shaped as a helix r ( t ) = h cos t, sin t,t i , t [0 , 2 ], if the density at every point is the square of the distance of the point to the origin....
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This note was uploaded on 10/06/2009 for the course MATH 445 taught by Professor Friedlander during the Fall '07 term at USC.
- Fall '07