This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 12 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 timeconsuming) 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....
View
Full
Document
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
 Friedlander
 Math

Click to edit the document details