Math 321 Vector and Complex Calculus for the Physical Sciences c Fabian Waleffe Department of Mathematics & Department of Engineering Physics University of Wisconsin, Madison 2016/1/18 ii \Any fool can know. The point is to understand." | Albert Einstein Contents 1 Vector Geometry and Algebra 1 1 Get your bearings . .1 1.1 Magnitude and direction . .1 1.2 Representations of 2D vectors . .2 1.3 Representations of 3D vectors . .3 2 Addition and scaling of vectors . .6 3 General Vector Spaces . .7 3.1 Axiomatic definition of a vector space . .7 3.2 The vector space Rn .....................7 4 Bases and Components . .8 5 Points and Coordinates . 11 5.1 Position vector . 11 5.2 Lines and Planes . 13 6 Dot Product . 14 7 Orthonormal bases . 18 8 Dot product and norm in Rn (Optional) . 20 9 Cross Product . 22 10 Index notation . 29 10.1 Levi-Civita symbol . 29 10.2 Einstein summation convention . 30 11 Mixed product and Determinant . 34 12 Points, Lines, Planes, etc. 39 13 Orthogonal Transformations and Matrices . 42 13.1 Change of cartesian basis . 42 13.2 Matrices . 46 13.3 Spherical coordinates with a twist: Euler Angles . 48 13.4 Determinant of a matrix (Optional) . 52 13.5 Three views of Ax = b (Optional) . 53 13.6 Eigenvalues and Eigenvectors (Math 320 not 321) . 55 2 Vector Calculus 57 1 Vector function of a scalar variable . 57 2 Motion of a particle . 59 3 Motion of a system of particles (optional) . 62 iii iv 4 Motion of a rigid body (optional) . 63 5 Curves . 66 5.1 Elementary curves . 66 5.2 Speeding through Curves . 67 5.3 Integrals along curves, or `line integrals' . 70 6 Surfaces . 73 6.1 Parameterizations . 73 6.2 Surface integrals . 80 6.3 Curves on surfaces . 81 7 Volumes . 82 8 Maps, curvilinear coordinates . 85 9 Change of variables . 88 10 Grad, div, curl . 93 10.1 Scalar fields and iso-contours . 93 10.2 Geometric concept of the Gradient . 93 10.3 Directional derivative, gradient and the r operator . 95 10.4 Div and Curl . 97 10.5 Vector identities . 97 10.6 Grad, Div, Curl in cylindrical and spherical coordinates . 99 11 Fundamental examples of vector fields . 103 12 Fundamental theorems of vector calculus . 104 12.1 Integration in R2 and R3 .................. 104 12.2 Fundamental theorem of Calculus . 105 12.3 Fundamental theorem in R2 ................. 105 12.4 Green and Stokes' theorems . 107 12.5 Divergence form of Green's theorem . 110 12.6 Gauss' theorem . 111 12.7 Other forms of the fundamental theorem in 3D . 113 3 Complex Calculus 117 1 Complex Numbers and Elementary functions . 117 1.1 Complex Algebra and Geometry . 117 1.2 Limits and Derivatives . 122 1.3 Geometric sums and series . 124 1.4 Ratio test . 125 1.5 Power series . 126 1.6 Complex transcendentals . 128 1.7 Polar representation . 131 1.8 Logs and complex powers . 132 2 Functions of a complex variable . 137 2.1 Visualization of complex functions . 137 2.2 Cauchy-Riemann equations . 138 2.3 Geometry of Cauchy-Riemann, Conformal Mapping . 142 2.4 Conformal mapping examples . 145 2.5 Joukowski Map . 149 3 Integration of Complex Functions . 151 c F. Waleffe, Math 321, 2016/1/18 v 3.1 Complex integrals are path integrals . 151 3.2 Cauchy's theorem . 152 3.3 Poles and Residues . 153 3.4 Cauchy's formula . 158 4 Real examples of complex integration . 161 Chapter 1 Vector Geometry and Algebra What is a vector? In calculus, you may have defined 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 re- duces 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 the observer and the system of coordinates. 1 Get your bearings 1.1 Magnitude and direction We write a = a a^ (1) for a vector a of magnitude jaj = a in the direction a^. The magnitude of vector a is denoted jaj which is a positive real number with appropriate physical units N (meters, Newtons, . ), jaj ≥ 0. The direction of vector a is denoted a^ which is a vector of unit magnitude, ja^j = 1. For that reason, direction vectors are often called unit vectors, but `unit' vectors have no physical units. For example, a ≡ 2 km heading 30◦clockwise from North a 30◦ specifies a horizontal displacement a of magnitude jaj = 2 kilometers and di- rection a^ ≡ 30◦ clockwise from North. −−! Two points A and B specify a displacement vector a = AB. The same −−! −−! displacement a starting from point C leads to point D, with CD = a = AB. Conversely, we can specify points (locations) by specifying displacements from 1 2 a reference point, thus point B is located displacement a from point A, and D D is a from C. B Vectors are denoted with a boldface: a, b, u, v, . , usually lower-case but some- a times upper-case as in a magnetic field B, electric field E or force F . Some authors write ~a for a and we use that notation to denote the displacement from a −−! C point A to point B as AB. Writing by hand, we use the typographical notation A for boldface which is a for a,a ^ for a^, etc. This notation allows distinguish- e e ing between a collection of unit vectors fa^1; a^2; a^3g ≡ fa^1,a ^2,a ^3g and the e e e components of a unit vector a^ ≡ a^ ≡ (^a1; a^2; a^3). e 1.2 Representations of 2D vectors For vectors in a plane (2D), a direction a^ can be specified by an angle from a reference direction as in the section 1.1 example. In navigation, that angle is usually defined clockwise from North and called an azimuthal angle or azimuth or heading. In mathematical physics, we specify the direction using an angle α counterclockwise from a reference direction x^. We can specify a 2D vector a by giving the pair (a; α) for vector a of magnitude jaj = a and direction a^ ≡ \angle α counterclockwise from x^." a a a ayy^ α x^ axx^ Figure 1.1: Polar and cartesian representations of vectors in a plane. We can also specify the vector a as a sum of displacements in two reference perpendicular directions, ax in direction x^ and ay in direction y^ perpendicular to x^. Now the pair (ax; ay) specifies a. The pairs (a; α) and (ax; ay) are equivalent representations of a, a ≡ (a; α) ≡ (ax; ay) but in general they are not equal (a; α) 6= (ax; ay): To express the equality of the representations, we need to include the direction vectors and write a = a a^ = ax x^ + ay y^ (2) then by Pythagoras and basic trigonometry ) ax = a cos α 2 2 2 , a = ax + ay; (3) ay = a sin α c F. Waleffe, Math 321, 2016/1/18 3 a^ = cos α x^ + sin α y^: (4) This confirms that a direction vector a^ is a vector of unit magnitude. The representation a = a a^, where a^ = a^(α) is a function of α, is a polar representation of the 2D vector a and a = axx^+ayy^ is a cartesian representation with x^ and y^ two arbitrary perpendicular directions. The vector a is specified by the pair (a; α) or (ax; ay), or many other pairs of numbers depending on how we specify the direction, for instance `20 feet toward the water tower then 5 meters toward the oak tree' specifies a displace- ment as (20ft,5m) but the reference directions may not be perpendicular. The same physical 2D vector can be represented by an infinity of pairs of numbers depending on how we choose our reference magnitudes and directions. Exercises: 1. If you move 2 miles heading 030◦ then 3 miles heading 290◦, how far are you from your original position? what heading from your original position? Make a sketch. All headings are measured clockwise from North. 2. If you move distance a heading α then distance b heading β, how far are you and in what heading from your original position? Make a sketch and show/explain your algorithm. Headings are clockwise from North. 3. Find jvj and v^ for the vector v ≡ −5 miles per hour heading Northeast. 4. True or False: in SI units ja^j = 1 meter while in CGS units ja^j = 1 cm. 1.3 Representations of 3D vectors How do you specify an arbitrary direction a^ in 3D space? Astronomers use an azimuth angle measured clockwise from North in the azelzen.gif 592×470 pixels 8/30/14 6:46 PM horizontal plane and an inclination (or zenith) angle measured from the vertical. An altitude (or elevation) angle measured up from the horizontal plane may be used instead of the inclination. Azimuth and elevation angles are also used in ballistics and computer graphics. The illustration is from NOAA, the National Ocean and Atmosphere Administration. Figure 1.2 shows the mathematical physics convention, where we use the angle β between a reference direction z^ (typically the vertical direction or the polar axis) and the arbitrary direction a^, as well as the angle α between the (z^; x^) and the (z^; a^) planes. Thus α is an azimuthal angle but defined to be posi- tive counterclockwise around z^ as opposed to the clockwise convention used in navigation.