2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4).

Transcription

1 Exam 3 Study Guide Math 223 Section 12 Fall 2015 Instructor: Dr. Gilbert 1. Which of the following vector fields are conservative? If you determine that a vector field is conservative, find a valid potential function for the vector field. (a) F = y sin zî + x sin zĵ + xy cos zˆk. (b) F = (y + z)î + zĵ + (x + y)ˆk. (c) F = e x cos yî e x cos yĵ + zˆk.

2 2 2. Evaluate C F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4). 3. Find the work done by the force F = xyî + (y x)ĵ on a particle which moves along a straight line from the point (1, 1) to (2, 3). 4. Let C be the curve given by r(t) = tî + (1 + t 3 )ĵ for 0 t 1. If F = x 3 y 4 î + x 4 y 3 ĵ, find F d r C in two ways: Using the parametrization, and then using the Fundamental Theorem of Calculus for Line Integrals.

3 5. Find a parametric representation for the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = Attempt to sketch the curve given by the parametric equations x = (4 + sin 20t) cos t, y = (4 + sin 20t) sin t, z = cos 20t. How would you describe this curve?

4 4 7. Sketch the curve given parametrically by r(t) = sin tî + 3ĵ + cos tˆk. Make sure to draw an arrow to show the orientation of the curve. 8. Find a parametrization for the helix which traces out a circle of radius 3 when projected onto the xy-plane, the rings of which are evenly spaced along the z-axis, begins at the point (3, 0, 0) and ends at the point (3, 0, 4π), after completing two full revolutions. 9. Using your knowledge of the helix, try and draw a sketch of the curve given parametrically by r(t) = e t cos 10tî + e t sin 10tĵ + e tˆk.

5 10. Find parametric equations for the tangent line to the curve given parametrically by x = cos t, y = 3e 2t, and z = 3e 2t at the point (1, 3, 3) Find the points of intersection of the tangent lines to the curve r(t) = sin(πt)î + 2 sin(πt)ĵ + cos(πt)ˆk at the points where t = 0 and t = 5.

6 6 12. The curves r 1 (t) = tî + t 2 ĵ + t 3ˆk and r2 (t) = sin tî + sin(2t)ĵ + tˆk intersect at the origin. Find their angle of intersection to the nearest degree.

7 13. A projectile is fired with an angle of elevation α and initial velocity v 0. Assuming that air resistance is negligible and the only external force is due to gravity, find the position function, r(t), of the projectile. What value of α maximizes the range (the horizontal distance traveled)? [Hint: Start with the fact that you know that a = gĵ, where g is the acceleration due to gravity.] 7

8 8 14. A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 50 above the horizontal. Is it a home run? (In other words, does the ball clear the fence?)

9 15. The position function of a particle is given by r(t) = t 2 î + 5tĵ + (t 2 16t)ˆk. When is the speed a minimum? Evaluate the line integral F d r, where F = yzî + xzĵ + xyˆk, and the path C is given by r(t) = tî + t 2 ĵ + t 3ˆk. C

10 Find the work done by the force field F = x 2 î + xyĵ on a particle that moves once around the circle x 2 + y 2 = 4 oriented in the counterclockwise direction. 18. Find the work done by the force field F = (y + z)î + (x + z)ĵ + (x + y)ˆk on a particle that moves along the line segment from (1, 0, 0) to (3, 4, 2).

11 19. Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire. Ampere s Law relates the electric current to its magnetic field and states that B d r = µ 0 I, C where I is the net current that passes through any surface bounded by a closed curve C and µ 0 is a constant called the permeability of free space. By taking C to be a circle with radius r, show that the magnitude B = B of the magnetic field at a distance r from the center of the wire is B = µ 0I 2πr. [Hint: use the fact that the line integral of any smooth vector field F over any piecewise smooth curve C can be written as F ˆT dr, C where at any point r, ˆT is a unit vector that lies tangent to the curve C at r, and dr represents an infinitesemal piece of arc length. 11

12 Show that the line integral C F d r is independent of path and evaluate the integral. (a) F = tan yî + x sec 2 yĵ, and C is any path from (1, 0) to (2, π/4). (b) F = (1 ye x )î + e x ĵ, and C is any path from (0, 1) to (1, 2).

13 (a) Suppose that F is an inverse square force field, that is, F = K r r 3, for some constant of proportionality K, where r = xî + yĵ + zˆk. Find the work done by F in moving an object from a point P along a path to a point Q in terms of the distances r 1 and r 2 from these point to the origin. (b) An example of an inverse square field is the electric field from an electric charge Q located at the origin: E = ɛqq r r 3, where ɛ is a constant which depends on the units, and q is the charge of a particle being affected by E. Suppose that an electron with a charge of C is located at the origin. A positive unit charge (q = 1 C) is positioned at a distance m from the electron and moves to a position half that distance from the electron. Use part (a) to find the work done by the electric field. (Use the value ɛ = )

14 Use Green s Theorem to evaluate the line integral along the given curve, oriented in such a way that the enclosed region lies to the left of the curve if we are traveling along the curve. (a) C (ey î + 2xe y ĵ) d r, where C is the square with sides x = 0, x = 1, y = 0, and y = 1. (b) C (x2 y 2 î + 4xy 3 ĵ) d r, where C is the triangle with vertices (0, 0), (1, 3), and (0, 3). (c) C ((y + e x )î + (2x + cos y 2 )ĵ) d r, where C is the boundary of the region enclosed by the parabolas y = x 2 and x = y 2.

15 (d) C (xe 2x î + (x 4 + 2x 2 y 2 )ĵ) d r, where C is the boundary of the region between the circles x 2 +y 2 = 1 and x 2 +y 2 = 4 (Note here that the outer circle will be oriented counterclockwise while the inner circle will be oriented clockwise). 15 (e) C (y3 î x 3 ĵ) d r, where C is the circle x 2 + y 2 = 4.

16 Use Green s Theorem to evaluate the theorem). C F d r (Check the orientation of the curve before applying (a) vecf = ( x + y 3 )î + (x 2 + y)ĵ, where C consists of the arc of the curve y = sin x from (0, 0) to (π, 0), followed by the line segment from (π, 0) to (0, 0). (b) F = (e x + x 2 y)î + (e y xy 2 )ĵ, where C is the circle x 2 + y 2 = 25, oriented clockwise.

17 24. If a circle C 1 with radius 1 rolls along the outside of the circle x 2 + y 2 = 16, a fixed point P on C 1 traces out a curve called an epicycloid, with parametric equations x = 5 cos t cos(5t), y = 5 sin t sin(5t). Graph the epicycloid and use Green s Theorem to find the area that it encloses. 17