Fall 2011 ME 566 Advanced Engineering Mathematics ENG ME 566 Advanced Engineering Mathematics Instructor: M. S. Howe EMA 218 (730 Commonwealth Ave) [email protected] Prerequisites: Multivariate Calculus; Ordinary Differential Equations; or instructor permission. It is expected that you can already: • solve simple first and second order linear ordinary differential equations • differentiate and integrate elementary functions, including trigonometric, exponential and hyperbolic functions • integrate by parts; evaluate simple surface and volume integrals • use the binomial theorem and the series expansions of elementary functions (sine, cosine, exponential, logarithmic and hyperbolic functions) • do all the problems in the prerequisites self test at the end of these notes. Course Outcomes: • consolidate understanding of vector calculus and applications to graduate level engi- neering • introductory understanding of complex variable theory with applications to engineering problems • ability to solve standard partial differential equations of engineering science using eigen- function expansions, Fourier transforms and generalised functions • become proficient in documenting calculations Textbook: Lectures are based on Mathematical Methods for Mechanical Sciences (M. S. Howe; 6th edition). It can be downloaded in pdf form from the ME 566 BlackBoard web site. You are expected to ‘read around’ the subject, and are recommended to consult other textbooks such as those listed on page 5. Course grading: • 4 take-home examinations (12.5% each) • final closed book examination (50%) Fall 2011 1 ME 566 Advanced Engineering Mathematics Fall 2011 ME 566 Advanced Engineering Mathematics Homework: Four ungraded homework assignments provide practice in applying techniques taught in class – model answers will be posted on BlackBoard. In addition there are four take home exams each consisting of a short essay and 5 problems. Each take-home examination problem is graded out of 10 as follows: up to 7 points for obtaining the correct answer and showing all your working; up to 3 points for setting-out the answer in a neat and readable manner. The essay is graded out of 25. You have at least ONE WEEK for each take-home examination; exam scripts are due at the end of class on the date specified. The following example illustrates how solutions should be set-out, and how you should include written explanations of your steps: · 1 3 3 3 Question: Use the divergence theorem to evaluate HS F dS, where F = 3(x ,y ,z ) and S is the surface of the sphere |x|≤R. Solution: To evaluate I = I F·dS S 1 3 3 3 | | where F = 3(x ,y ,z ) and S is the surface x = R. By the divergence theorem I = Z div F dV, V where the integration is over the volume V of the sphere, and ∂F ∂F ∂F ! 1 div F = 1 + 2 + 3 = (3x2 +3y2 +3z2)=r2,r= qx2 + y2 + z2. ∂x ∂y ∂z 3 Using spherical polar coordinates, and taking dV =4πr2dr (because the integrand is radially symmetric): R 4π R I = Z r2dV = Z 4πr4dr = r5 r<R 0 5 0 4π .˙.I= R5 5 Fall 2011 2 ME 566 Advanced Engineering Mathematics Fall 2011 ME 566 Advanced Engineering Mathematics SAMPLE ESSAY Essay: Write a short essay (not more than two sides of paper) in which you give the definition of a vector field, define the curl of a vector field F(x), and state and prove Stokes’ theorem. A single valued vector function defined over a region of space is called a vector field. For example, the velocity vector in a fluid, and vector gravitational and electrical force distributions are vector fields. The ‘curl’ of a vector field F(x) is a new vector field curl F whose value at x is determined by the following limiting procedure. For any fixed unit vector ˆa , the component of curl F in the direction of ˆa is 1 ˆa · curl F = lim I F·dr (1) A→0 A C where C is a closed contour enclosing x in the plane through x with normal ˆa . C encloses an area A of the plane, and is traversed in the positive sense with respect to the direction ˆa . For rectangular coordinates x =(x, y, z), with unit vectors i, j, k respectively parallel to the x, y, and z directions, this definition yields ijk ∂ ∂ ∂ curl F ≡∇×F = ∂x ∂y ∂z F F F 1 2 3 ∂F ∂F ! ∂F ∂F ! ∂F ∂F ! = 3 − 2 i + 1 − 3 j + 2 − 1 k, ∂y ∂z ∂z ∂x ∂x ∂y where F =(F1,F2,F3). The definition (1) also leads to a direct proof of Stokes’ theorem I F·dr = Z curl F · dS ≡ Z n · curl FdS, C S S where C is a closed contour, S an open, two-sided surface bounded by C, and the line integral is taken along C in the positive direction with respect to the unit normal n on S. We can write H F·dr = lim →∞ H F·dr, where the surface S is partitioned into C k Pk Ck infinitesimal elements of area δSk with unit normal nk and boundary Ck, because the integral along each section of Ck common to two adjacent elements enters twice, once for each element, but with opposite signs, and therefore makes no contribution to the sum. What remains is just the integral along those sections of the Ck that make up the contour C. The definition (1), with aˆ = nk, then gives I I Z F·dr = lim X F·dr = lim X nk · curl FδSk = curl F · dS. Q.E.D. k→∞ k→∞ C k Ck k S Fall 2011 3 ME 566 Advanced Engineering Mathematics Fall 2011 ME 566 Advanced Engineering Mathematics Schedule Sep T6 Sep R8 Sep T13 Vector/Multivariate Calculus Sep R15 Sep T20 Sep R22 TH #1 Sep T27 Sep R29 TH #1 due Oct T4 Oct R6 Complex Variable and Integration Oct T11 Oct R13 Oct T18 TH #2 Oct R20 Oct T25 TH #2 due Oct R27 Nov T1 Nov R3 PDEs and Fourier transforms Nov T8 Nov R10 TH #3 Nov T15 Nov R17 TH #3 due Nov T22 Nov T29 TH #4 Dec R1 Review Dec T6 TH #4 due Dec R8 Fall 2011 4 ME 566 Advanced Engineering Mathematics Fall 2011 ME 566 Advanced Engineering Mathematics Syllabus During the course of the semester you are expected to read and understand the material in those sections of MMMS, 6th edition, indicated after each heading in the following syllabus: 1. Vector/Multivariate Calculus Sections 2.1 - 2.5; Section 2.7. Assignment 1: Take-home examination 1; Essay 1. 2. Complex Variable and Integration Sections 3.1 - 3.10. Assignment 2: Take-home examination 2; Essay 2. 3/4. Partial differential equations; Fourier series and transforms Sections 1.9, 1.10, 1.11; Sections 4.1 - 4.3, 4.6, 4.9, 4.10. Assignment 3: Take-home examination 3; Essay 3. Assignment 4: Take-home examination 4; Essay 4. Alternative Texts: (not recommended for purchase): Lopez, R. J. Advanced Engineering Mathematics. Addison Wesley. Kreyszig, E. Advanced Engineering Mathematics. Wiley. Greenberg, M. E. Advanced Engineering Mathematics. Prentice Hall. Zill, D. G. and Cullen, M. R. Advanced Engineering Mathematics. Jones and Bertlett. Fall 2011 5 ME 566 Advanced Engineering Mathematics Fall 2011 ME 566 Advanced Engineering Mathematics ASSIGNMENT 1 Problems 2A 1. Find the value of x given that a =3i − 2j and b =4i + xj are perpendicular. [6]. 3. Solve for x the vector equation x + a(b · x)=c, where a, b, c are constant vectors. What happens when a · b = −1? [x = c − a(b · c)/(1 + a · b)]. 5. Solve the simultaneous equations x + y × p = a, y + x × p = b. [x = {(p · a)p + a − b × p}/(1 + p2), y = {(p · b)p + b − a × p}/(1 + p2)]. 7. Show that (a × b) · (a × c) × d =(a · d)(a · b × c). Problems 2B Calculate the gradients of 1. ϕ = x [i]. 3. ϕ = rn, r = xi + yj + zk. [nrn−2r]. 5. ϕ = r·∇(x + y + z), r = xi + yj + zk. [i + j + k]. Find the directional derivatives in the direction of a of √ 7. ϕ =ex cos y, a =(2, 3, 0), x =(2,π,0) [−2e2/ 13]. Find the unit normal to the surfaces: √ 9. z = px2 + y2 at x =(3, 4, 5) [(3, 4, −5)/5 2]. Problems 2C Find the divergence of 1. F = yi + zj + xk [0]. 3. F =(x, y2,z3)[1+2y +3z2]. 5. F = xyz(i + j + k)[yz + xz + xy]. 7. F = r(r · a), r = xi + yj + zk, a = constant. [4r · a]. Prove that 9. ∇2(rn)=n(n +1)rn−2, r = xi + yj + zk. 11. div(f∇g) − div(g∇f)=f∇2g − g∇2f. 1 ·∇ 2 13. V = 6 HS n (r )dS, where r = xi + yj + zk and V is the volume enclosed by S. · Evaluate HS n FdS when 2 − 2 1 15. F =(x, x y, x z) and S is the surface of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) (0, 0, 1) [ 6 ]. | | 4 17. F = axi + byj + czk where a, b, c are constants, and S is the unit sphere x =1 [3 π(a + b + c)]. Fall 2011 6 ME 566 Advanced Engineering Mathematics Fall 2011 ME 566 Advanced Engineering Mathematics Problems 2E · Evaluate the line integrals RC F dr: 4 6 2 2 − −192 1. F =(3x , 3y , 0) where C is the curve: x + y =4,z= 0 from (2, 0, 0) to ( 2, 0, 0) [ 5 ]. x 4y/x 2z/y 2 3 3 4 3 2 − 13 3. F =(e , e , e ) where C is r =(t, t ,t ), 0 <t<1[8 e + 4 e +e 8 ].