PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Summary...1 3. Assignments...2
i PMTH212, Multivariable Calculus Assignment Summary 2010 Assignment Date to be Posted Description Number 1 23 February Problems 2 2 March Problems 3 9 March Problems 4 16 March Problems 5 23 March Problems 6 30 March Problems 7 27 April Problems 8 4 May Problems 9 11 May Problems 10 18 May Problems 11 25 May Problems 12 1 June Problems This assignment summary may be detached for your notice board.
1 PURE MATHEMATICS 212 Multivariable Calculus Summary Please refer to the Unit Information and Assessment booklets for detail information. Lecture Notes and Reference Book Lecture notes will be provided. The reference book is Calculus by H. Anton, John Wiley & Sons. Assessment The 12 assignments contribute 30% of the final assessment. The student must submit reasonable attempts on at least 8 assignments. The final examination is compulsory. It contributes 70% of the final assessment. The student must achieve at least 40% in the examination. Examination The examination will be of three hours duration. You are allowed to bring with you three A4 sized pieces of paper of hand-written notes. Both sides can be used. No printed and photocopied materials are allowed. Unit Coordinator Dr. Shusen Yan Phone: (02) 6773 3154 Email: syan@turing.une.edu.au Fax: 6773 3312
2 ASSIGNMENT 1 1. Show that (4, 5, 2), (1, 7, 3) and (2, 4, 5) are vertices of an equilateral triangle. 2. Find the distance from the point (5, 2, 3) to the (a) yz-plane; (b) z-axis; (c) x-axis. 3. Find the terminal point of v =< 7, 6 > if the initial point is (2, 1). 4. Find the orthogonal projection of u on a. (a) u = 2i + j, a = 3i + 2j; (b) u = 7i + j + 3k, a = 5i + k. 5. Find a vector n which is perpendicular to the plane determined by the points A(0, 2, 1),B(1, 1, 2), and C( 1, 2, 0). 6. Find parametric equations for the line in R 2 through (0, 3) parallel to the line x = 5 + t,y = 1 2t. 7. Determine whether the planes are parallel or perpendicular. (a) plane x y + 3z 2 = 0 and plane 3x z = 1; (b) plane 3x 2y + z = 1 and plane 4x + 3y 2z = 4; (c) plane y = 4x 2z + 3 and plane x = 1y + 1z. 4 2 8. Find an equation of the plane: (a) through the points P 1 ( 2, 1, 4),P 2 (1, 0, 3) and perpendicular to the plane 4x y + 3z = 2; (b) that contains the point (2, 0, 3) and the line x = 1 + t, y = t, z = 4 + 2t.
3 ASSIGNMENT 2 1. Name and sketch the surface: x y 2 4z 2 = 0. 2. Find an equation of the orthogonal projection onto the xy-plane of the curve of intersection of the following surfaces. Identify the curve. (a) The cone z 2 = x 2 + y 2 and the plane z = 2y. (b) The ellipsoid x 2 + 2y 2 + z 2 = 2 and the plane z = x. 3. Describe the graph of the vector equations: (a) r = 3i + 2 cos tj + 2 sin tk; (b) r = 2i + (t 2 + t)j + tk. 4. Determine whether r = sin(πt)i + (2t ln t)j + (t 2 t)k is a smooth function of the parameter t. 5. Let u,v,w be differentiable vector-valued functions of t. Prove that d du dw [u [v w)] = [v w] + u [dv w] + u [v dt dt dt dt ]. 6. (a) Evaluate < te t, ln t > dt. (b) Find the arc length of the curve: x = 1 2 t, y = 1 3 (1 t)3/2, z = 1 3 (1 + t)3/2, 1 t 1.
4 ASSIGNMENT 3 1. Find the unit tangent vector T and the unit normal vector N for the given value of t: x = sint, y = cos t, z = 1 2 t2 ; t = 0. 2. Find the curvature at the indicated point: r(t) = e t cos ti + e t sin tj + e t k; t = 0. 3. Show that for a plane curve described by r = x(t)i + y(t)j, the curvature κ(t) is κ(t) = x y y x (x 2 + y 2 ) 3/2, where a prime denotes differentiation with respect to t. 4. Let f(x,y) = xy + 3. Find (a) f(x + y,x y); (b) f(xy, 3x 2 y 3 ). 5. Let g(x,y) = ye 3x, x(t) = ln(t 2 + 1), y(t) = t. Find g(x(t),y(t)). 6. Find g(u(x,y,z), v(x,y,z), w(x,y,z)), if g(x,y,z) = z sin xy, u(x,y,z) = x 2 z 3, v(x,y,z) = πxyz, and w(x,y,z) = xy/z. 7. Describe the level surfaces. (a) f(x,y,z) = 3x y + 4z 1; (b) f(x,y,z) = z x 2 z 2.
5 ASSIGNMENT 4 1. Find the region where the function f is continuous. (a) f(x,y) = ln(1 2x 2 y 2 ) (b) f(x,y) = cos( xy ) 1+x 2 +y 2 2. Find the limit, if it exists. (a) (b) x 3 + 16y 4 lim (x,y) (0,0) x 2 + 4y 2 1 cos(x 2 + y 2 ) lim (x,y) (0,0) x 2 + y 2 3. Show xy (a) x 2 + y can be made to approach any value in the interval 2 [ 1, 1 ] by letting 2 2 (x,y) (0, 0) along an appropriate straight line y = mx; (b) lim (x,y) (0,0) xy x 2 +y 2 does not exist. 4. Find f x, f y and f z for the following functions: f(x,y,z) = ( xz 1 x 2 y 2 ) 3/4. 5. (a) Show that f(x,y) = e x sin y + e y cos x satisfies Laplace s Equation 2 f/ x 2 + 2 f/ y 2 = 0. (b) Show that u(x,y) = ln(x 2 + y 2 ) and v(x,y) = 2 tan 1 ( y ) satisfy the Cauchyx Riemann Equations u/ x = v/ y, u/ y = v/ x. 6. Let f(x,y) = (x 3 + y 3 ) 1/3. (a) Show that f y (0, 0) = 1; (b) At what points, if any, does f y (x,y) fail to exist?
6 ASSIGNMENT 5 1. Let f(x,y) = sin(x 2 + y 3 ). Find f xy and f yx and verify their equality. 2. Let z = x 2 y tanx, x = u/v, y = u 2 v 2. Find z/ v and z/ u by the chain rule. 3. Show that if u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations (see Assignment 4, Question 5), and if x = r cosθ and y = r sinθ, then u r = 1 v r θ, v r = 1 u r θ. 4. Find equations for the tangent plane and the normal line to the given surfaces at the point P for: x 2 y 4z 2 = 7, P( 3, 1, 2). 5. (a) Find a unit vector in the direction in which f(x,y) = x y decreases most x+y rapidly at P(3, 1); and find the rate of change of f at P in that direction. (b) Find the directional derivative of f(x,y) = e x sec y at P(0,π/4) in the direction from this point to the origin.
7 ASSIGNMENT 6 1. Find a unit vector u that is perpendicular at P(0, 3, 0) to the level surface of f(x,y,z) = (x 3y + 4z) 1/2 through P. 2. Let f be a function of one variable and let z = f(x + 2y). Show that 2 z/ x z/ y = 0. 3. Assume that F(x,y,z) = 0 defines z implicitly as a function of x and y. Show that z/ x = F/ x, z/ y = F/ y F/ z F/ z. 4. Use the formulas in Question 4 to find z/ x and z/ y for (a) x 2 3yz 2 + xyz 2 = 0; (b) ln(1 + z) + xy 2 + z = 1.
8 ASSIGNMENT 7 1. Locate all relative maxima, relative minima and saddle points: f(x,y) = xy x 3 y 2. 2. Find the absolute extrema of the given function on the indicated closed and bounded set R: f(x,y) = x 2 3y 2 2x + 6y; R is the square region with vertices (0,0), (0,2), (2,2) and (2,0). 3. Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint: f(x,y) = 4x 3 + y 2 ; 2x 2 + y 2 = 1 4. Find the point on the plane x + 2y + z = 1 that is closest to the origin. 5. Find the points on the surface xy z 2 = 1 that are closest to the point P(0, 0, 0).
9 ASSIGNMENT 8 1. Evaluate the iterated integral: 1 x 2 1 x 2(x2 y)dy dx. 2. Evaluate the double integral. (a) R (x2 xy)da; R is the region enclosed by y = x and y = 3x x 2 ; (b) R x(1 + y2 ) 1/2 da; R is the region in the first quadrant enclosed by y = x 2,y = 4 and x = 0. 3. Evaluate the following integrals by first reversing the order of integration: (a) (b) 1 4 0 4x 1 π/2 0 sin 1 y e y2 dy dx; sec 2 (cos x)dxdy. 4. (a) Use polar coordinates to evaluate R 1 1 + x 2 + y 2 da, where R is the sector in the first quadrant that is bounded by y = 0,y = x and x 2 + y 2 = 4. (b) Evaluate the iterated integral by converting to polar coordinates. 1 1 y 2 0 0 cos(x 2 + y 2 )dxdy. 5. Find the surface area of the portion of the cone z = x 2 + y 2 that lies inside the cylinder x 2 + y 2 = 2x.
10 ASSIGNMENT 9 1. Evaluate the triple integral ydv, G where G is the solid enclosed by the plane z = y, the xy-plane, and the parabolic cylinder y = 1 x 2. 2. Express the integral as an equivalent integral in which the z-integration is performed first, the y-integration second and the x-integration last: 3 (9 z 2 ) 1/2 (9 y 2 z 2 ) 1/2 0 0 0 f(x,y,z)dxdy dz 3. Let G be the rectangular box defined by a x b,c y d,k z l. Show that [ ] [ b ] [ d ] l f(x)g(y)h(z)dv = f(x)dx g(y) dy h(z) dz. G a 4. (a) Use cylindrical coordinates to find the volume of the solid that is enclosed between the surface r 2 + z 2 = 20 and the surface z = r 2. (b) Use spherical coordinates to find the volume of the solid within the cone φ = π/4 and between the spheres ρ = 1 and ρ = 2. 5. Let G be the solid in the first octant bounded by the sphere x 2 + y 2 + z 2 = 4 and the coordinate planes. Evaluate. G x 2 z dv c k
11 ASSIGNMENT 10 1. Let F(x,y) = (3x + 2y)i + (2x y)j. Evaluate C F dr where C is (a) the curve y = x 3 from (0,0) to (1,1); (b) the curve x = y 2 from (0,0) to (1,1); 2. Find the work done if a particle moves along the curve x = t,y = 1/t, 1 t 3 where subject to the force F(x,y) = (x 2 + xy)i + (y x 2 y)j. 3. Show that (1,3) ( 1,2) y 2 dx + 2xy dy is independent of the path, and evaluate the integral by (a) using Theorem 1 in Lecture 28; (b) integrating along the line segment from ( 1, 2) to (1, 3). 4. Show that the integral (3,3) (1,1) (e x ln y ey )dx + (ex x y ey ln x)dy is independent of the path, and find its value by any method. 5. Use Green s Theorem to calculate y tan 2 xdx + tanxdy, where C is the circle x 2 + (y + 1) 2 = 1. C
12 ASSIGNMENT 11 1. Evaluate the surface integral σ (x + y)ds where σ is the portion of the plane z = 6 2x 3y in the first octant. 2. Evaluate the surface integral σ (x2 + y 2 )ds, where σ is the portion of the sphere x 2 + y 2 + z 2 = 4 above the plane z = 1. 3. Evaluate σ F nds where F(x,y,z) = x2 i + yxj + zxk and σ is the portion of the plane 6x + 3y + 2z = 6 in the first octant, oriented by unit normals with positive components. 4. Use the Divergence Theorem to evaluate σ F nds, where n is the outer unit normal to σ. (a) F(x,y,z)=xi+yj+zk, σ is the surface of the solid bounded by the paraboloid z = 1 x 2 y 2 and the xy-plane. (b) F(x,y,z) = (x 3 e y )i (y 3 +sinz)j+(z 3 xy)k, where σ is the surface of the solid bounded by z = 4 x 2 y 2 and the xy-plane. [Hint: Use spherical coordinates.]
13 ASSIGNMENT 12 1. Use Stokes Theorem to evaluate C F dr, where F(x,y,z) = xzi + 3x 2 y 2 j + yxk; C is the rectangle in the plane z = y with vertices (0,0,0), (1,0,0), (1,3,3) and (0,3,3), oriented in that order. 2. Show that if the components of F and their first- and second-order partial derivatives are continuous, then div(curlf)=0. 3. Find the net volume of the fluid that crosses the surface σ in the direction of the orientation in one unit of time where the flow field is given by F(x,y,z) = xi + yj + 2zk, σ is the portion of the cone z 2 = x 2 + y 2 between the planes z = 1 and z = 2, oriented by upward unit normals.