Mathematics 317 Assignment #10: due Wednesday, April 7, 2010 1. Verify Green's theorem when F = y 2 i + xj and C is the circle x 2 + y 2 = 1. 2. Verify Green's theorem when F = ( x 2 - y 2 )i + ( x 3 + y 3 ) j and R is the rectangular region given by 1 x
Mathematics 317T2 Assignment #6 due Wednesday, March 3, 2010 1. Suppose the force field F(x, y, z) is conservative in the domain D. If ( x, y, z ) is a potential for F, show that the lines of force for F are perpendicular to the equipotential surfaces = c
Mathematics 317 Assignment #3: due Wednesday, January 27, 2010 1. Neptune's period is 164.8 years. Show that it is about 30 times as far from the sun as earth is. 2. Suppose a particle has an elliptical orbit under a force per unit mass / r 2 , i.e., towa
Mathematics 317 T2 Assignment #7 due Wednesday, March 10, 2010 1. For any two twice differentiable functions f ( x, y, z ) and g ( x, y, z ), show that (a) ( fg ) = f 2 g + f g. (b) ( fg - gf ) = f 2 g - g 2 f . (c) ( fg + gf ) = 2 ( fg ). (d) ( fg ) = f
Mathematics 317T2 Assignment #9 due Wednesday, March 25, 2010 1. Verify the divergence theorem when F = xi + yj + zk and is the closed surface bounded by the cylindrical surface x 2 + y 2 = 1 and the planes z = 0, z = 1. 2. Suppose E is conservative in th
Mathematics 317 T2 Assignment #5 due Friday, February 12, 2010 1. If F = (2 x + y )i + (3 x - 2 y ) j, evaluate F dr where C is:
C
(a) The straight line path from (0,0) to (1,1). (b) The parabolic path y = x 2 from (0,0) to (1,1). 2. Let F = xyi - yj + zk
Mathematics 317 Assignment #4: due Friday, February 5, 2010 1. The velocity field of a fluid is given by q = - xi + yj. Sketch the velocity field and find the equation of the streamline passing through the point (1, 1).
2. Sketch the field lines for the f
MATHEMATICS 317 Assignment #2 due at the beginning of class on Wednesday, January 20th 1 (a) For a curve defined by f ( x, y ) = 0, show that the radius of curvature R at any
point of the curve is given by the expression R =
[ f x + f y ]3 / 2
2 2
2 f x f
Mathematics 317 Solutions to Midterm #1 Fall 2009
1. (a)
dr ds & v (t ) = r = = vT. ds dt
(b) Differentiating (a) with respect to t, one obtains dT ds 2 dT 2 & & & & & & a = v = vT + vT = vT + v = vT + v = vT + (t )v N, ds dt ds from the first Frenet-Ser
Mathematics 317 Solutions to Assignment #6 1. F = . Hence a line of force lies in the direction of . But equipotential surface = const. Hence lines of force are perpendicular to equipotential surfaces.
2. (a) Clearly any vector field of the form F = a ( x
Mathematics 317: Solutions to Assignment #3 1. Let T1 ,T2 be the respective periods for the orbits of Neptune and Earth and let a1 , a 2 be the respective lengths of the semi-major axes of the elliptical orbits of Neptune and Earth. Then a1 , a 2 can appr
Mathematics 317 Solutions to Ass. #2 1. (a) The equation f ( x, y ) = 0 defines y as an implicit function of x. From the
equation, one has
df dx
= f x + f y y = 0
=
d2 f dx 2
f y3
= f xx + 2 f xy y + f yy y 2 + f y y = 0. Hence
. As shown in class, the r
MATHEMATICS 317 Solutions to Ass. #1 1(a) & & & r (t ) = x(t )i + y (t ) j; r (t ) = x(t )i + y (t ) j = xi - 2 / x 2 j & x = x (1)
& y = -2 / x 2
( 2) 1 -2t e + b, i.e., a2
Eqn. (1) x = ae t [from Eqn. (2) ] y =
1 r (t ) = ae t i + 2 e - 2t + b j for un
Mathematics 317 Midterm #3: Friday, November 20, 2009 1. Answer Question #1. Do only one of Questions #2 and #3. Each question is of equal value. No notes, books or calculators are permitted. 2. Time Limit: 50 minutes 1. Consider the vector field given by
Mathematics 317 Midterm #2: Monday, November 2, 2009 1. Do only two questions. Each question is of equal value. If you attempt more than two questions, only the first two will be marked. 2. No notes, books or calculators are permitted. 3. Time Limit: 50 m
Mathematics 317 Midterm #1: Monday, October 5, 2009 1. Do only two questions. Each question is of equal value. If you attempt more than two questions, only the first two will be marked. 2. No notes, books or calculators are permitted. A data sheet is incl
MATHEMATICS 317 Assignment #1 due at the beginning of class on Wednesday, January 13th
& 1. A particle moves in the xy-plane with velocity v = r = dr / dt = xi - (2 / x 2 ) j. (a) Find the position of the particle, i.e., r(t) at time t, if r(0) = i + j. (
MATH 317 T2 SOLUTIONS TO MIDTERM #1 1. As shown in class, the curvature (t ) =
| r (t ) r (t ) | . In this question | r (t ) | 3 r (t ) = f (t )i + g (t ) j r (t ) = f (t )i + g (t ) j, r (t ) = f (t )i + g (t ) j. Hence
. | r (t ) r (t ) |=| f (t ) g (t
Math 317 Solutions to Assignment #5 1. (a) C can be parametrized by y = x, 0 x 1. Along C, dy = dx. Hence
1 1
F dr = [2 x + x) + (3x - 2 x)]dx = 4 xdx = 2.
C 0 0
(b) Here C can be parametrized by x = t , y = t 2 , 0 t 1. Along C, dx = dt , dy = d (t 2 )