6. Vector Integral Calculus in Space

Save this PDF as:

Size: px
Start display at page:

Transcription

1 6. Vector Integral alculus in pace 6A. Vector Fields in pace 6A-1 Describegeometricallythefollowingvectorfields: a) xi +yj +zk ρ b) xi zk 6A-2 Write down the vector field where each vector runs from (x,y,z) to a point half-way towards the origin. 6A-3 Write down the velocity field F representing a rotation about the x-axis in the direction given by the right-hand rule (thumb pointing in positive x-direction), and having constant angular velocity ω. 6A-4 Write down the most general vector field all of whose vectors are parallel to the plane 3x 4y +z = 2. 6B. urface Integrals and Flux 6B-1 Without calculating, find the flux of xi +yj +zk through the sphere of radius a and center at the origin. Take n pointing outward. 6B-2 Without calculation, find the flux of k through the infinite cylinder x 2 + y 2 = 1. (Take n pointing outward.) 6B-3 Without calculation, find the flux of i through that portion of the plane x+y+z = 1 lying in the first octant (take n pointed away from the origin). 6B-4 Find F d, where F = yj, and = the half of the sphere x2 +y 2 +z 2 = a 2 for which y 0, oriented so that n points away from the origin. 6B-5 Find F d, where where F = zk, and is the surface of Exercise 6B-3 above. 6B-6 Find F d, where F = xi + yj + zk, and is the part of the paraboloid z = x 2 +y 2 lying underneath the plane z = 1, with n pointing generally upwards. Explain geometrically why your answer is negative. 6B-7* Find xi +yj +zk F d, where F = x 2 +y 2, and is the surface of Exercise 6B-2. +z2 6B-8 Find F d, where F = yj and is that portion of the cylinder x2 +y 2 = a 2 between the planes z = 0 and z = h, and to the right of the xz-plane; n points outwards. 6B-9* Find the center of gravity of a hemispherical shell of radius a. (Assume the density is 1, and place it so its base is on the xy-plane. 6B-10* Let be that portion of the plane 12x+4y+3z = 12 projecting vertically onto the plane region (x 1) 2 +y 2 4. Evaluate a) the area of b) zd c) (x 2 +y 2 +3z)d 1

2 2 E EXERIE 6B-11* Let be that portion of the cylinder x 2 + y 2 = a 2 bounded below by the xy-plane and above by the cone z = (x a) 2 +y 2. a) Find the area of. Recall that 1 cosθ = 2sin(θ/2). (Hint: remember that the upper limit of integration for the z-integral will be a function of θ determined by the intersection of the two surfaces.) b) Find the moment of inertia of about the z-axis. There should be nothing to calculate once you ve done part (a). c) Evaluate z 2 d. 6B-12 Find the average height above the xy-plane of a point chosen at random on the surface of the hemisphere x 2 +y 2 +z 2 = a 2, z Divergence Theorem 6-1 alculate div F for each of the following fields a) x 2 yi +xyj +xzk b)* 3x 2 yzi +x 3 zj +x 3 yk c)* sin 3 xi +3ycos 3 xj +2xk 6-2 alculate div F if F = ρ n (xi +yj +zk), and tell for what value(s) of n we have div F = 0. (Use ρ x = x/ρ, etc.) 6-3 Verify the divergence theorem when F = xi +yj +zk and is the surface composed of the upper half of the sphere of radius a and center at the origin, together with the circular disc in the xy-plane centered at the origin and of radius a. 6-4* Verify the divergence theorem if F is as in Exercise 3 and is the surface of the unit cube having diagonally opposite vertices at (0,0,0) and (1,1,1), with three sides in the coordinate planes. (All the surface integrals are easy and do not require any formulas.) 6-5 By using the divergence theorem, evaluate the surface integral giving the flux of F = xi +z 2 j +y 2 k over the tetrahedron with vertices at the origin and the three points on the positive coordinate axes at distance 1 from the origin. 6-6 Evaluate F d over the closed surface formed below by a piece of the cone z 2 = x 2 + y 2 and above by a circular disc in the plane z = 1; take F to be the field of Exercise 6B-5; use the divergence theorem. 6-7 Verify the divergence theorem when is the closed surface having for its sides a portion of the cylinder x 2 + y 2 = 1 and for its top and bottom circular portions of the planes z = 1 and z = 0; take F to be a) x 2 i +xyj b)* zyk c)* x 2 i +xyj +zyk (use (a) and (b)) 6-8 uppose div F = 0 and 1 and 2 are the upper and lower hemispheres of the unit sphere centered at the origin. Direct both hemispheres so that the unit normal is up, i.e., has positive k-component. a) how that F d = F d, and interpret this physically in terms of flux. 1 2 b) tate a generalization to an arbitrary closed surface and a field F such that div F = 0.

3 6. VETOR INTEGRAL ALULU IN PAE 3 6-9* Let F be the vector field for which all vectors are aimed radially away from the origin, with magnitude 1/ρ 2. a) What is the domain of F? b) how that div F = 0. c) Evaluate F d, where is a sphere of radius a centered at the origin. Does the fact that the answer is not zero contradict the divergence theorem? Explain. d) Prove using the divergence theorem that F d over a positively oriented closed surface has the value zero if the surface does not enclose the origin, and the value 4π if it does. (F is the vector field for the flow arising from a source of strength 4π at the origin.) 6-10 A flow field F is said to be incompressible if. Assume that F is continuously differentiable. how that F d = 0 for all closed surfaces F is the field of an incompressible flow div F = how that the flux of the position vector F = xi + yj + zk outward through a closed surface is three times the volume contained in that surface. 6D. Line Integrals in pace 6D-1 Evaluate F dr for the following fields F and curves : a) F = yi +zj xk; is the twisted cubic curve x = t, y = t 2, z = t 3 running from (0,0,0) to (1,1,1). b) F is the field of (a); is the line running from (0,0,0) to (1,1,1) c) F is the field of (a); is the path made up of the succession of line segments running from (0,0,0) to (1,0,0) to (1,1,0) to (1,1,1). d) F = zxi + zyj + xk; is the helix x = cost, y = sint, z = t, running from (1,0,0) to (1,0,2π). 6D-2 Let F = xi +yj +zk; show that F dr = 0 for any curve lying on a sphere of radius a centered at the origin. 6D-3* a) Let be the directed line segment running from P to Q, and let F be a constant vector field. how that F dr = F PQ. b) Let be a closed space polygon P 1 P 2...P n P 1, and let F be a constant vector field. how that F dr = 0. (Use part (a).) c) Let be a closed space curve, F a constant vector field. how that F dr = 0. (Use part (b).)

4 4 E EXERIE 6D-4 a) Let f(x,y,z) = x 2 +y 2 +z 2 ; calculate F = f. b) Let be the helix of 6D-1d above, but running from t = 0 to t = 2nπ. alculate the work done by F moving a unit point mass along ; use three methods: (i) directly (ii) by using the path-independence of the integral to replace by a simpler path (iii) by using the first fundamental theorem for line integrals. 6D-5 Let F = f, where f(x,y,z) = sin(xyz). What is the maximum value of F dr over all possible paths? Give a path for which this maximum value is attained. 1 6D-6* Let F = f, where f(x,y,z) =. Find the work done by F carrying x+y +z +1 a unit point mass from the origin out to along a ray in the first octant. (Take the ray to be x = at,y = bt,z = ct, with a, b, c positive and t 0.) 6E. Gradient Fields in pace 6E-1 Which of the following differentials are exact? For each one which is, express it in the form df for a suitable function f(x,y,z), using one of the systematic methods. a) x 2 dx+y 2 dy +z 2 dz b) y 2 zdx+2xyzdy +xy 2 dz c) y(6x 2 +z)dx+x(2x 2 +z)dy +xydz 6E-2 Find curl F, if F = x 2 yi +yzj +xyz 2 k. 6E-3 The fields F below are defined for all x,y,z. For each, a) show that curl F = 0; b) find a potential function f(x,y,z), using either method, or inspection. (i) xi +yj +zk (ii) (2xy +z)i +x 2 j +xk (iii) yzi +xzj +xyk 6E-4 how that if f(x,y,z) and g(x,y,z) are two functions having the same gradient, then f = g +c for some constant c. (Use the Fundamental Theorem for Line Integrals.) 6E-5 For what values of a and b will F = yz 2 i + (xz 2 + ayz)j + (bxyz + y 2 )k be a conservative field? Using these values, find the corresponding potential function f(x, y, z) by one of the systematic methods. 6E-6 a) Define what it means for Mdx+Ndy +Pdz to be an exact differential. will be exact. b) Find all values of a,b,c for which (axyz +y 3 z 2 )dx+(a/2)x 2 z +3xy 2 z 2 +byz 3 )dy +(3x 2 y +cxy 3 z +6y 2 z 2 )dz c) For those values of a,b,c, express the differential as df for a suitable f(x,y,z).

5 6. VETOR INTEGRAL ALULU IN PAE 5 6F. tokes Theorem 6F-1 Verify tokes theorem when is the upper hemisphere of the sphere of radius one centered at the origin and is its boundary; i.e., calculate both integrals in the theorem and show they are equal. Do this for the vector fields a) F = xi +yj +zk; b) F = yi +zj +xk. 6F-2 Verify tokes theorem if F = yi + zj + xk and is the portion of the plane x+y +z = 0 cut out by the cylinder x 2 +y 2 = 1, and is its boundary (an ellipse). 6F-3 Verify tokes theorem when is the rectangle with vertices at (0,0,0), (1,1,0), (0,0,1), and (1,1,1), and F = yzi +xzj +xyk. 6F-4* Let F = M i +N j +P k, where M,N,P have continuous second partial derivatives. a) how by direct calculation that div(curl F) = 0. b) Using (a), show that F nd = 0 for any closed surface. 6F-5 Let be the surface formed by the cylinder x 2 + y 2 = a 2, 0 z h, together with the circular disc forming its top, oriented so the normal vector points up or out. Let F = yi +xj +x 2 k. Find the flux of F through (a) directly, by calculating two surface integrals; (b) by using tokes theorem. 6G-1 Which regions are simply-connected? 6G. Topological Questions a) first octant b) exterior of a torus c) region between two concentric spheres d) three-space with one of the following removed: i) a line ii) a point iii) a circle iv) the letter H v) the letter R vi) a ray 6G-2 how that the fields F = ρ n (xi +yj +zk), where ρ = x 2 +y 2 +z 2, are gradient fields for any value of the integer n. (Use ρ x = x/ρ, etc.) Then, findthepotentialfunctionf(x,y,z). (Itiseasiesttophrasethequestionintermsof differentials: one wants df = ρ n (xdx+ydy+zdz); for n = 0, you can find f by inspection; from this you can guess the answer for n 0 as well. The case n = 2 is an exception, and must be handled separately. The printed solutions use this method, somewhat more formally phrased using the fundamental theorem of line integrals.) 6G-3* If D is taken to be the exterior of the wire link shown, then the little closed curve cannot be shrunk to a point without leaving D, i.e., without crossing the link. Nonetheless, show that is the boundary of a two-sided surface lying entirely inside D. (o if F is a field in D such that curl F = 0, the above considerations show that F dr = 0.)

6 6 E EXERIE 6G-4* In cylindrical coordinates r,θ,z, let F = ϕ, where ϕ = tan 1 a) Interpret ϕ geometrically. What is the domain of F? z r 1. b) From the geometric interpretation what will be the value of F dr around a closed path that links with the unit circle in the xy-plane (for example, take to be the circle in the yz-plane with radius 1 and center at (0,1,0)? 6H. Applications to Physics 6H-1 Prove that F = 0. What are the appropriate hypotheses about the field F? 6H-2 how that for any closed surface, and continuously differentiable vector field F, curl F d = 0. Do it two ways: a) using the divergence theorem; 6H-3* Prove each of the following ( φ is a (scalar) function): a) (φf) = φ F+F φ b) (φf) = φ F+( φ) F c) (F G) = G F F G b) using tokes theorem. 6H-4* The normal derivative. If is an oriented surface with unit normal vector n, and φ is a function defined and differentiable on some domain containing, then the normal derivative of φ on is defined to be the directional derivative of φ in the direction n. In symbols (on the left is the notation for the normal derivative): φ n = φ n. Prove that if is closed and D its interior, and if φ has continuous second derivatives inside D, then φ n d = 2 φdv. (This shows for example that if you are trying to find a harmonic function φ defined in D and having a prescribed normal derivative on, you must be sure that φ has been n φ prescribed so that d = 0. n 6H-5* Formulate and prove the analogue of the preceding exercise for the plane. 6H-6* Prove that, if is a closed surface with interior D, and φ has continuous second derivatives in D, then φ φ n d = φ( 2 φ)+( φ) 2 dv. 6H-7* Formulate and prove the analogue of the preceding exercise for a plane. D D

7 6. VETOR INTEGRAL ALULU IN PAE 7 6H-8 A boundary value problem.* uppose you want to find a function φ defined in a domain containing a closed surface and its interior D, such that (i) φ is harmonic in D and (ii) φ = 0 on. a) how that the two conditions imply that φ = 0 on all of D. (Use Exercise 6.) b) Instead of assuming (ii), assume instead that the values of φ on are prescribed as some continuous function on. Prove that if a function φ exists which is harmonic in D and has these prescribed boundary values, then it is unique there is only one such function. (In other words, the values of a harmonic function function on the boundary surface determine its values everywhere inside.) (Hint: Assume there are two such functions and consider their difference.) 6H-9 Vector potential* In the same way that F = φ F = 0 has the partial converse F = 0 in a simply-connected region F = f, so the theorem F = G F = 0 has the partial converse (*) F = 0 in a suitable region F = G, for some G. G is called a vector potential for F. A suitable region is one with this property: whenever P lies in the region, the whole line segment joining P to the origin lies in the region. (Instead of the origin, one could use some other fixed point.) For instance, a sphere, a cube, or all of 3-space would be suitable regions. uppose for instance that F = 0 in all of 3-space. Then G exists in all of 3-space, and is given by the formula (**) G = 1 0 tf(tx,ty,tz) Rdt, R = xi +yj +zk The integral means: integrate separately each component of the vector function occurring in the integrand, and you ll get the corresponding component of G. We shall not prove this formula here; the proof depends on Leibniz rule for differentiating under an integral sign. We can however try out the formula. a) Let F = yi +zj +xk. heck that div F=0, find G from the formula (**), and check your answer by verifying that F = curl G. b) how that G is unique up to the addition of an arbitrary gradient field; i.e., if G is one such field, then all others are of the form (***) G = G+ f, for an arbitrary function f(x,y,z). (how that if G has the form (***), then F = curl G ; then show conversely that if G is a field such that curl G = F, then G has the form (***).)

8 8 E EXERIE 6H-10 Let B be a magnetic field produced by a moving electric field E. Assume there are no charges in the region. Then one of Maxwell s equations in differential form reads B = 1 c E t. What is the integrated form of this law? Prove your answer, as in the notes; you can assume that the partial differentiation can be moved outside of the integral sign. 6H-11* In the preceding problem if we also allow for a field j which gives the current density at each point of space, we get Ampere s law in differential form (as modified by Maxwell): B = 1 ( 4πj + E ). c t Give the integrated form of this law, and deduce it from the differential form, as done in the notes.

9 18.02 Notes and Exercises by A. Mattuck and Bjorn Poonen with the assistance of T.hifrin and. LeDuc c M.I.T

Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x

18.02 Multivariable Calculus Fall 2007

MIT OpenourseWare http://ocw.mit.edu 18.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Lecture 30. Tue, Nov

4B. Line Integrals in the Plane

4. Line Integrals in the Plane 4A. Plane Vector Fields 4A-1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region

Answers and Solutions to Section 13.7 Homework Problems 1 19 (odd) S. F. Ellermeyer April 23, 2004

Answers and olutions to ection 1.7 Homework Problems 1 19 (odd). F. Ellermeyer April 2, 24 1. The hemisphere and the paraboloid both have the same boundary curve, the circle x 2 y 2 4. Therefore, by tokes

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem

alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use

Math Review for Exam 3

1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)

Integral Theorems. September 14, We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative,

Integral Theorems eptember 14, 215 1 Integral of the gradient We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative, F (b F (a f (x provided f (x

(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3

Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those

18.02 Multivariable Calculus Fall 2007

MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. V9. Surface Integrals Surface

Vector Calculus, Maths II

Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent

Peter Alfeld Math , Fall 2005

WeBWorK assignment due 9/2/05 at :59 PM..( pt) Consider the parametric equation x = 2(cosθ + θsinθ) y = 2(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 7 6 π? 2.( pt) Let a = (-2 4 2) and

29.3. Integral Vector Theorems. Introduction. Prerequisites. Learning Outcomes

Integral ector Theorems 9. Introduction arious theorems exist relating integrals involving vectors. Those involving line, surface and volume integrals are introduced here. They are the multivariable calculus

18.02 Multivariable Calculus Fall 2007

MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 4. Line Integrals in the

Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VectorCalculus1 due 05/03/2008 at 02:00am EDT.

Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment Vectoralculus due 05/03/008 at 0:00am EDT.. ( pt) rochesterlibrary/setvectoralculus/ur V.pg onsider the transformation T : x = 35 35 37u

ES.182A Topic 44 Notes Jeremy Orloff

E.182A Topic 44 Notes Jeremy Orloff 44 urface integrals and flux Note: Much of these notes are taken directly from the upplementary Notes V8, V9 by Arthur Mattuck. urface integrals are another natural

ARNOLD PIZER rochester problib from CVS Summer 2003

ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.

Ma 1c Practical - Solutions to Homework Set 7

Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut

Math 11 Fall 2016 Final Practice Problem Solutions

Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11

1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv

1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N

ES.182A Topic 45 Notes Jeremy Orloff

E.8A Topic 45 Notes Jeremy Orloff 45 More surface integrals; divergence theorem Note: Much of these notes are taken directly from the upplementary Notes V0 by Arthur Mattuck. 45. Closed urfaces A closed

The Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)

Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green

Math 23b Practice Final Summer 2011

Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order

Stokes Theorem. MATH 311, Calculus III. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Stokes Theorem

tokes Theorem MATH 311, alculus III J. Robert Buchanan Department of Mathematics ummer 2011 Background (1 of 2) Recall: Green s Theorem, M(x, y) dx + N(x, y) dy = R ( N x M ) da y where is a piecewise

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS

MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,

4. Line Integrals in the Plane

4. Line Integrals in the Plane 4A. Plane Vector Fields 4A- a) All vectors in the field are identical; continuously differentiable everywhere. b) The vector at P has its tail at P and head at the origin;

Tom Robbins WW Prob Lib1 Math , Fall 2001

Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment due 9/7/0 at 6:00 AM..( pt) A child walks due east on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 7 miles

MATHS 267 Answers to Stokes Practice Dr. Jones

MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

Solutions for the Practice Final - Math 23B, 2016

olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

Sept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.

MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign

Line, surface and volume integrals

www.thestudycampus.com Line, surface and volume integrals In the previous chapter we encountered continuously varying scalar and vector fields and discussed the action of various differential operators

53. Flux Integrals. Here, R is the region over which the double integral is evaluated.

53. Flux Integrals Let be an orientable surface within 3. An orientable surface, roughly speaking, is one with two distinct sides. At any point on an orientable surface, there exists two normal vectors,

Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Hilary Term 211: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things that are (and are not) vectors. Differentiation and integration

S12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)

OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

Math 11 Fall 2007 Practice Problem Solutions

Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

Review problems for the final exam Calculus III Fall 2003

Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)

Line and Surface Integrals. Stokes and Divergence Theorems

Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function

Assignment 11 Solutions

. Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which

The Divergence Theorem

Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points

MATH2000 Flux integrals and Gauss divergence theorem (solutions)

DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

MATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.

MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c

Math Exam IV - Fall 2011

Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and

Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals

Math 280 Calculus III Chapter 16 Sections: 16.1, 16.2 16.3 16.4 16.5 Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1

Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent

Name: Instructor: Lecture time: TA: Section time:

Math 222 Final May 11, 29 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 1 problems on 16 pages worth a total of 2 points. Look over your test package right

Keble College - Hilary 2015 CP3&4: Mathematical methods I&II Tutorial 4 - Vector calculus and multiple integrals II

Keble ollege - Hilary 2015 P3&4: Mathematical methods I&II Tutorial 4 - Vector calculus and multiple integrals II Tomi Johnson 1 Prepare full solutions to the problems with a self assessment of your progress

Mathematics (Course B) Lent Term 2005 Examples Sheet 2

N12d Natural Sciences, Part IA Dr M. G. Worster Mathematics (Course B) Lent Term 2005 Examples Sheet 2 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damtp.cam.ac.uk. Note that

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A

A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART

Math 5BI: Problem Set 9 Integral Theorems of Vector Calculus

Math 5BI: Problem et 9 Integral Theorems of Vector Calculus June 2, 2010 A. ivergence and Curl The gradient operator = i + y j + z k operates not only on scalar-valued functions f, yielding the gradient

Solutions to Sample Questions for Final Exam

olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the

LINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4

LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts

ENGI 4430 Gauss & Stokes Theorems; Potentials Page 10.01

ENGI 443 Gauss & tokes heorems; Potentials Page.. Gauss Divergence heorem Let be a piecewise-smooth closed surface enclosing a volume in vector field. hen the net flux of F out of is F d F d, N 3 and let

Math 31CH - Spring Final Exam

Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate

f dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1

hapter 6 Integrals In this chapter we will look at integrals in more detail. We will look at integrals along a curve, and multi-dimensional integrals in 2 or more dimensions. In physics we use these integrals

Print Your Name: Your Section: Mathematics 1c. Practice Final Solutions This exam has ten questions. J. Marsden You may take four hours; there is no credit for overtime work No aids (including notes, books,

Final Exam Review Sheet : Comments and Selected Solutions

MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,

MATH 52 FINAL EXAM SOLUTIONS

MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

MATH20411 PDEs and Vector Calculus B

MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411

LINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4

LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts

EE2007: Engineering Mathematics II Vector Calculus

EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 EE2007:

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line

is the curve of intersection of the plane y z 2 and the cylinder x oriented counterclockwise when viewed from above.

The questions below are representative or actual questions that have appeared on final eams in Math from pring 009 to present. The questions below are in no particular order. There are tpicall 10 questions

Review Sheet for the Final

Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence

MA227 Surface Integrals

MA7 urface Integrals Parametrically Defined urfaces We discussed earlier the concept of fx,y,zds where is given by z x,y.wehad fds fx,y,x,y1 x y 1 da R where R is the projection of onto the x,y - plane.

McGill University April 16, Advanced Calculus for Engineers

McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer

Practice problems. m zδdv. In our case, we can cancel δ and have z =

Practice problems 1. Consider a right circular cone of uniform density. The height is H. Let s say the distance of the centroid to the base is d. What is the value d/h? We can create a coordinate system

MAT 211 Final Exam. Spring Jennings. Show your work!

MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin

Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)

eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your

Integral Vector Calculus

ontents 29 Integral Vector alculus 29.1 Line Integrals Involving Vectors 2 29.2 Surface and Volume Integrals 34 29.3 Integral Vector Theorems 54 Learning outcomes In this Workbook you will learn how to

SURFACE INTEGRALS Question 1 Find the area of the plane with equation x + 3y + 6z = 60, 0 x 4, 0 y 6. 8 Question A surface has Cartesian equation y z x + + = 1. 4 5 Determine the area of the surface which

MLC Practice Final Exam

Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C

Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line

GRIET(AUTONOMOU) POPULAR QUETION IN ADVANED ALULU UNIT-. If u = f(e z, e z, e u u u ) then prove that. z. If z u, Prove that u u u. zz. If r r e cos, e sin then show that r u u e [ urr u ]. 4. Find J,

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

MATH 2400 Final Exam Review Solutions

MATH Final Eam eview olutions. Find an equation for the collection of points that are equidistant to A, 5, ) and B6,, ). AP BP + ) + y 5) + z ) 6) y ) + z + ) + + + y y + 5 + z 6z + 9 + 6 + y y + + z +

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute

alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In

Practice problems **********************************************************

Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without

14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Iterated Integrals. Iterated Integrals. MAC2313 Calculus III - Chapter 14

14 Multiple Integration 14.1 Iterated Integrals and Area in the Plane Objectives Evaluate an iterated integral. Use an iterated integral to find the area of a plane region. Copyright Cengage Learning.

18.02 Multivariable Calculus Fall 2007

MIT OpenourseWare http://ocw.mit.edu 18.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Lecture 21. Test for

Vector Calculus. Dr. D. Sukumar. February 1, 2016

Vector Calculus Dr. D. Sukumar February 1, 2016 Green s Theorem Tangent form or Ciculation-Curl form c Mdx + Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form Stoke s Theorem

Ma 227 Final Exam Solutions 12/22/09

Ma 7 Final Exam Solutions //9 Name: ID: Lecture Section: Problem a) (3 points) Does the following system of equations have a unique solution or an infinite set of solutions or no solution? Find any solutions.

The Divergence Theorem

The Divergence Theorem 5-3-8 The Divergence Theorem relates flux of a vector field through the boundary of a region to a triple integral over the region. In particular, let F be a vector field, and let