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Math 321 Vector and Complex Calculus for the Physical Sciences ©Fabian Walee Department of Mathematics & Department of Engineering Physics University of Wisconsin, Madison

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Math 321 Vector and Complex Calculus for the Physical Sciences ©Fabian Walee Department of Mathematics & Department of Engineering Physics University of Wisconsin, Madison January 25, 20182 �... �F. Walee, Math 321, January 25, 2018 TERMS OF USE: These notes and other materials (homework, exams, solutions) are provided for the private use of current and past Math 321 students at UW-Madison. You should not redistribute any of these materials or post them onto external websites such as coursehero.com, or any other such websites. We are providing many ways to get help on the material. In addition to these comprehensive notes, we oer interactive lectures, discussion sections and oce hours by faculty and TAs as well as interactive course websites such as piazza.com, where students can ask and answer questions, including anonymously if they prefer.Preface These notes have been developed to ll the gap between the mathematics covered in standard multivariable calculus courses and the vector and complex calculus needed for intermediate and advanced undergraduate courses in the physical and engineering sciences. Emphasis is placed on geometric understanding of the fundamental concepts and connections with physical world experiences and applications. Vector, index and matrix notations are discussed since each have their own merits and limitations and all three are used interchangeably in applications. The mathematical concepts and tools covered here are fundamental to sciences such as mechanics, electromagnetism, uid dynamics, aerodynamics, transport phenomena, ight dynamics, astrodynamics, continuum mechanics, elasticity, plasma physics, geophysical and astrophysical uid dynamics, for example, as well as computer graphics that requires a deeper understanding of the mathematical modeling of curves, surfaces and volumes. It is common for instructors of courses in those areas to spend 1/3 or more of their time quickly ‘reviewing’ the requisite vector and complex calculus that is needed for adequate presentation and understanding of the material. This creates excessive redundancy at an introductory level that is frustrating and inecient for students and instructors alike. The material is presented at an intermediate to advanced undergraduate level. Students are expected to have mastered basic geometry, trigonometry, algebra and calculus including multi-variable calculus (partial derivatives, multiple integrals). Familiarity with dierential equations and linear algebra is recommended but not perhaps not required. Knowledge of introductory physics is assumed. The concepts are motivated, derived and justied carefully but the approach is intended to be more ‘intuitive’ than ‘rigorous,’ although those terms are subjective. The derivations and proofs should be studied and understood as they lead to deeper understanding of the concepts. Solved examples are provided but not too many as to drown out the fundamental concepts. The concepts and examples should be studied thoroughly before attempting the exercises. Exercises appear after most subsections to encourage the productive cycle of studying a particular concept then practicing its applications in non-trivial problems and going back to studying the concept more deeply. Learning is an iterative process. Acknowledgements: Thanks to Prof. Jean-Luc Thieault and Drs. Anakewit ‘Tete’ Boonkasame and Alfredo Wetzel for using and commenting on these notes. Skull surface image courtesy of https://doc.cgal.org/latest/Surface_mesher/ index.html. 34 ©F. Walee, Math 321, January 25, 2018Contents 1 Vector Geometry and Algebra 9 1 Find your bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1 Magnitude and Direction . . . . . . . . . . . . . . . . . . . . . 9 1.2 Vectors on a plane . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Vectors in space . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Addition and scaling of vectors . . . . . . . . . . . . . . . . . 14 2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Abstract Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The vector space Rn . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Bases and Components . . . . . . . . . . . . . . . . . . . . . . 18 3 Points and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 Position vector . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . 22 3.3 Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Medians of a triangle . . . . . . . . . . . . . . . . . . . . . . . 24 4 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1 Geometry and algebra of dot product . . . . . . . . . . . . . . 26 4.2 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Dot product and norm in Rn (Optional) . . . . . . . . . . . . 33 5 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 Geometry and algebra of cross product . . . . . . . . . . . . . 35 5.2 Double cross product (‘Triple vector product’) . . . . . . . . . 37 5.3 Orientation of Bases . . . . . . . . . . . . . . . . . . . . . . . 38 6 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1 Levi-Civita symbol . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Einstein summation convention . . . . . . . . . . . . . . . . . 44 7 Mixed product and Determinant . . . . . . . . . . . . . . . . . . . . . 48 8 Points, Lines, Planes, etc. . . . . . . . . . . . . . . . . . . . . . . . . . 53 9 Orthogonal Transformations and Matrices . . . . . . . . . . . . . . . 56 9.1 Change of cartesian basis in 2D . . . . . . . . . . . . . . . . . 56 9.2 Change of cartesian basis in 3D . . . . . . . . . . . . . . . . . 57 9.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9.4 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 9.5 Determinant of a matrix (Optional) . . . . . . . . . . . . . . . 69 56 ©F. Walee, Math 321, January 25, 2018 9.6 Three views of Ax = b (Optional) . . . . . . . . . . . . . . . 69 9.7 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.8 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 72 2 Vector Calculus 73 1 Vector functions and their derivatives . . . . . . . . . . . . . . . . . . 73 2 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 Motion of a single particle . . . . . . . . . . . . . . . . . . . . 79 2.2 Motion of a system of particles (optional) . . . . . . . . . . . 83 2.3 Motion of a rigid body (optional) . . . . . . . . . . . . . . . . 84 3 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1 Elementary curves . . . . . . . . . . . . . . . . . . . . . . . . 88 3.2 Speeding through Curves . . . . . . . . . . . . . . . . . . . . 90 3.3 Integrals over curves . . . . . . . . . . . . . . . . . . . . . . . 92 4 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1 Parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2 Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3 Curves on surfaces . . . . . . . . . . . . . . . . . . . . . . . . 105 5 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Maps, curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . 108 7 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8 Grad, div, curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.1 Scalar elds and iso-contours . . . . . . . . . . . . . . . . . . 117 8.2 Geometric concept of the Gradient . . . . . . . . . . . . . . . 117 8.3 Directional derivative, gradient and the r operator . . . . . . 119 8.4 Div and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.5 Vector identities . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.6 Grad, Div, Curl in cylindrical and spherical coordinates . . . . 125 9 Inverting Grad, Div, Curl (optional) . . . . . . . . . . . . . . . . . . . 130 10 Fundamental theorems of vector calculus . . . . . . . . . . . . . . . . 131 10.1 Integration in R2 and R3 . . . . . . . . . . . . . . . . . . . . . 131 10.2 Fundamental theorem of Calculus . . . . . . . . . . . . . . . . 132 10.3 Fundamental theorem in R2 . . . . . . . . . . . . . . . . . . . 132 10.4 Green and Stokes’ theorems . . . . . . . . . . . . . . . . . . . 134 10.5 Divergence form of Green’s theorem . . . . . . . . . . . . . . 137 10.6 Gauss’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 138 10.7 Other forms of the fundamental theorem in 3D . . . . . . . . 140 3 Complex Calculus 143 1 Complex Numbers and Elementary functions . . . . . . . . . . . . . . 143 1.1 Complex Algebra and Geometry . . . . . . . . . . . . . . . . 143 1.2 Limits and Derivatives . . . . . . . . . . . . . . . . . . . . . . 148 1.3 Geometric sums and series . . . . . . . . . . . . . . . . . . . . 149 1.4 Ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 1.5 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1.6 Complex transcendentals . . . . . . . . . . . . . . . . . . . . 154©F. Walee, Math 321, January 25, 2018 7 1.7 Polar representation . . . . . . . . . . . . . . . . . . . . . . . 158 1.8 Logs and complex powers . . . . . . . . . . . . . . . . . . . . 159 2 Functions of a complex variable . . . . . . . . . . . . . . . . . . . . . 164 2.1 Visualization of complex functions . . . . . . . . . . . . . . . 164 2.2 Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . 165 2.3 Geometry of Cauchy-Riemann, Conformal Mapping . . . . . 170 2.4 Conformal mapping examples . . . . . . . . . . . . . . . . . . 174 2.5 Joukowski Map . . . . . . . . . . . . . . . . . . . . . . . . . . 176 3 Integration of Complex Functions . . . . . . . . . . . . . . . . . . . . 179 3.1 Complex integrals are path integrals . . . . . . . . . . . . . . 179 3.2 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 181 3.3 Poles and Residues . . . . . . . . . . . . . . . . . . . . . . . . 181 3.4 Cauchy’s formula . . . . . . . . . . . . . . . . . . . . . . . . . 186 4 Real examples of complex integration . . . . . . . . . . . . . . . . . . 1908 ©F. Walee, Math 321, January 25, 2018Chapter 1 Vector Geometry and Algebra 1 Find your bearings What is a vector? In calculus, you may have dened vectors as lists of numbers such as (a1; a2) or (a1; a2; a3). Mathematicians favor that approach nowadays because it readily generalizes to n dimensional space (a1; a2; : : : ; an) and reduces geometry to arithmetic. But in physics, and in geometry before that, you encountered vectors as quantities with both magnitude and direction such as displacements, velocities and forces. The geometric point of view emphasizes the invariance of these quantities with respect to changes of the system of coordinates. 1.1 Magnitude and Direction The prototypical vectors are the displacements in two or three dimensional space such as the displacement of a boat on the surface of a lake, or a drone ying over a prairie. We write a = a a^ (1) for a vector a of magnitude a in the direction a^. N α a a Fig. 1.1: Vector a with magnitude a, heading α In navigation, the direction a^ is specied by the heading (or azimuth) α which is the clockwise angle from the Northern direction. The magnitude a is specied in appropriate physical units such as meters for a displacement, or meters/second for a velocity. Magnitude does not have to come before direction, ‘1km Northeast’ is the same as ‘Northeast 1km,’ a a^ = a^ a: Vectors are denoted in boldface type such as a, b, u, v, . . . , usually lower-case but often upper-case also in Physics, for example for the magnetic eld B, electric eld E or resultant force F . The magnitude of vector a is written in regular type a or with vertical bars jaj jaj = jaa^j = a ≥ 0: (2 [Show More]

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