ECE 206: Advanced Calculus 2 Department of Electrical and Computer Engineering University of Waterloo Fall 2014 Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 1/39
Course details Instructor Dr. Oleg Michailovich ( olegm@uwaterloo.ca , EIT 4127, ext. 38247) Office hours TBD in according to the class and instructor’s preferences Course websites ece.uwaterloo.ca/~ece206 (general information, lecture notes & slides, laboratories, supporting material, etc.) learn.uwaterloo.ca/ (lab and graded assignments, exams and marks) Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 2/39
Marking scheme Laboratory assignments (6 assignments × 3%): 18% Graded assignments (6 assignments × 2%): 12% Midterm exam: 20% Final exam: 50% Solutions for the laboratories may be submitted in pairs while graded assignments must be completed individually. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 3/39
Recommended course textbook Michael D. Greenberg, Advanced Engineering Mathematics , 2nd ed. However .... The class material will be mainly based on the lecture notes by Prof. Andrew Heunis which are available at the course website. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 4/39
Course outline 1 Multi-dimensional integration Two-dimensional integration Three-dimensional integration 2 Scalar and vector fields Motivating examples Definition of scalar and vector fields 3 Curves and paths in space Motivating examples Paths and parametric representation of curves Derivatives along a path and tangent to a curve Simple curves and closed curves 4 Line integral and arc length Line integral of a vector field Line integral of a scalar field and arc length Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 5/39
Course outline (cont.) 5 Conservative vector fields Gradient of a scalar field Conservative vector fields Conservation of energy 6 Green’s theorem in the plane Green’s theorem for rectangles Green’s theorem: the general case 7 Surfaces, surface areas and surface integrals Parametric representation of surfaces Tangents to a surface and smooth surfaces Area of a surface Surface integral of a scalar field Surface integral of a vector field 8 Vector calculus The divergence, Laplacian, and curl differential operators Theorem of Stokes Divergence theorem of Gauss-Ostogradskii The continuity equation Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 6/39
Course outline (cont.) 9 The basic laws of electricity and magnetism Static electric fields Static magnetic fields Time-varying fields 10 Maxwell’s equations The Ampere-Maxwell law for time-varying fields Maxwell’s equations Electromagnetic waves without sources Electromagnetic waves with sources 11 Cylindrical and spherical coordinates Polar coordinates Cylindrical coordinates Spherical coordinates Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 7/39
Questions? Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 8/39
General overview The course is about vector calculus and the calculus of complex variables . The three pillars of vector calculus are: Green’s theorem, Stokes’ theorem and the Gauss-Ostrogradskii theorem. Used in electromagnetism, aerodynamics, fluid mechanics, classi- cal mechanics, quantum mechanics and gravitational physics. Allows representing the main laws of electricity and magnetism as a set of just four equations, called Maxwell’s equations . Richard P. Feynman: “From a long view of the history of mankind - seen, from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics”. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 9/39
Maxwell’s marvellous equations For given electric field E , magnetic field B , charge density ρ , and current density field J , Maxwell’s equations ∇ · E = ρ ǫ 0 ∇ · B = 0 ∇ × E + ∂ B ∂t = 0 ∂ E ∇ × B = µ 0 J + ǫ 0 µ 0 ∂t So what are the symbols ∇· and ∇× standing for? Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 10/39
Two Dimensional Integration We want to integrate a real-valued function f : D → R where D ⊂ R 2 is a rectangular domain defined as � ( x, y ) ∈ R 2 | a ≤ x ≤ b, c ≤ y ≤ d � D = = [ a, b ] × [ c, d ] . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 11/39
Finite partition Subdivide the intervals [ a, b ] and [ c, d ] into n + 1 equally spaced points { x i } n i =0 and { y j } n j =0 , respectively, such that a = x 0 < x 1 < . . . < x n = b, c = y 0 < y 1 < . . . < y n = d, with spacings ∆ x := x i +1 − x i = b − a ∆ y := y j +1 − y j = d − c , . n n Let D i,j to be the small rectangular defined by ( x, y ) ∈ R 2 | x i ≤ x ≤ x i +1 , y j ≤ y ≤ y j +1 � � D i,j = = [ x i , x i +1 ] × [ y j , y j +1 ] and let r i , j = ( ξ i , η j ) ∈ D i,j , which implies x i ≤ ξ i ≤ x i +1 y j ≤ η j ≤ y j +1 Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 12/39
Riemann sum Definition We define the Riemann sum of the function f on the rectangle D as n n � � S n := f ( ξ i , η j )∆ x ∆ y, i =0 j =0 for any n ∈ { 1 , 2 , . . . } Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 13/39
Riemann integral Definition If the sequence of Riemann sums { S n , n = 1 , 2 , ... } converges to a limit S as n → ∞ , and the limit S is the same for every choice of points ( ξ i , η j ) ∈ D i,j , then S is called the (Riemann) integral of the function f over the rectangle D . Various notations for the integral S are � � � � f ( x, y ) dxdy, f ( x, y ) dA, fdxdy, fdA. D D D D Remarks: If the Riemann sums do not converge to any limit, the integral is said to be undefined . Fortunately, we need never be concerned with this fact, for the class of functions which can be integrated over D is simply huge. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 14/39
Fubini’s theorem If we fix some x ∈ [ a, b ], a function f ( x, y ) depends only on y in the interval c ≤ y ≤ d . For any such x , let h 1 ( x ) be defined as: � d h 1 ( x ) := f ( x, y ) dy ∀ a ≤ x ≤ b. c In exactly the same way we also define the function h 2 ( y ) as: � b h 2 ( y ) := f ( x, y ) dx ∀ c ≤ y ≤ d. a Fubini’s theorem � b � d � f ( x, y ) dxdy = h 1 ( x ) dx = h 2 ( y ) dy. D a c Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 15/39
Fubini’s theorem (cont.) A more detailed way to formulate Fubini’s theorem is: �� d � �� b � � b � d � f ( x, y ) dxdy = f ( x, y ) dy dx = f ( x, y ) dx dy. D a c c a Fubini’s theorem reduces evaluation of an integral over a rectang- le to the successive evaluations of two integrals over intervals . These are called iterated integrals . Both choices will work but in practice it is often the case that one choice involves less work than the other. Example: Integrate f ( x, y ) = x 2 + y 2 over D := [ − 1 , 1] × [0 , 1]. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 16/39
Integrals over non-rectangular domains What if D ⊂ R 2 is not a rectangular domain in the x − y plane. To define such D suppose that φ 1 : [ a, b ] → R and φ 2 : [ a, b ] → R are given continuous functions over some fixed interval a ≤ x ≤ b such that φ 1 ( x ) ≤ φ 2 ( x ) , ∀ a ≤ x ≤ y. Definition The region D ⊂ R 2 is called y -simple with lower function φ 1 ( x ), upper function φ 2 ( x ) and common interval of definition a ≤ x ≤ b , when � ( x, y ) ∈ R 2 | a ≤ x ≤ b, � D = φ 1 ( x ) ≤ y ≤ φ 2 ( x ) . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 17/39
Non-rectangular domains (cont.) Let the constants c and d be defined as: c < φ 1 ( x ) ≤ φ 2 ( x ) < d, ∀ a ≤ x ≤ y. Then, D is contained within E (i.e., D ⊆ E ), where � ( x, y ) ∈ R 2 | a ≤ x ≤ b, c ≤ y ≤ d � E = = [ a, b ] × [ c, d ] . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 18/39
Non-rectangular domains (cont.) Now define the function f ∗ : E → R 2 as given by � f ( x, y ) , if ( x, y ) ∈ D f ∗ ( x, y ) = 0 , if ( x, y ) ∈ E \ D It is then evident that � � f ( x, y ) dxdy = f ∗ ( x, y ) dxdy D E Now, by Fubini’s theorem �� d � � b � f ∗ ( x, y ) dxdy = f ∗ ( x, y ) dy dx = E a c �� φ 2 ( x ) � �� φ 2 ( x ) � � b � b = f ∗ ( x, y ) dy dx = f ( x, y ) dy dx. φ 1 ( x ) φ 1 ( x ) a a Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 19/39
Fubini’s theorem for y -simple regions Fubini’s theorem for y -simple regions Suppose that D is any y -simple region with lower function φ 1 ( x ), upper function φ 2 ( x ) and common interval of definition a ≤ x ≤ b , and f : D → R is a given function. Then �� φ 2 ( x ) � � b � f ( x, y ) dxdy = f ( x, y ) dy dx. D a φ 1 ( x ) Example: For φ 1 ( x ) = 0, φ 2 ( x ) = √ 1 + cos x , and x ∈ [0 , 2 π ], find the integral of f ( x, y ) = 2 y . � D x 2 y dxdy , where D is a triangular Example: Evaluate the integral area bounded by the lines x = 0, y = 0, and x + y = 1. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 20/39
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