Math and Physics RefresherGeometryCalculusApproximationsDifferential EquationsComplex NumbersVectors (see also Appendix H in HRK, vol.2)Vector CalculusPhysics – MechanicsNewton’s Laws:Conservation of Momentum:Work and Energy:Conservation of Energy:Special RelativityCircular motion:PHY2061 Enriched Physics 2 Lecture Notes RefresherMath and Physics RefresherThis course assumes that you have studied Newtonian mechanics in a previous calculus-based physics course (i.e. PHY2060) and at least have co-registered in a vector calculus course (Calc 3). Listed below are some of the concepts in basic math, calculus, and physics that you are expected to know or to acquire during this course. This is not a complete summary of introductory math and physics. It is only meant to be a refresher ofsome of the concepts used in this course. Please report any inaccuracies to the professor.Geometry1. Pythagorean Theorem:The square of the hypotenuse of a right triangle is the sum of the squares of the two legs: c a b2 2 2 2. Circumference of a circle: C R23. Volume of a sphere: V R4334. Surface area of sphere:S R42Calculus1. Differentiation:You are expected to be able to take simple derivatives:ddxx nxddxx xddxx xddxe en nx x 1sin coscos sin2. Product Ruleddxf x g x f xdgdxg xdfdxddxx x x x x( ) ( ) ( ) ( )sin cos sin Example: a fD. Acosta Page 1 1/14/2019 c a bPHY2061 Enriched Physics 2 Lecture Notes Refresher3. Chain Ruleddxf g xdfdgdgdxddxx xddxx x xafb gc h c h Examples: sin cosexp exp2 2 222 24. IntegrationYou are expected to be able to perform simple integrals:x dxnxdx x xdx x xdx e en nx xzzzz111cos sinsin cos A purist would note that a constant should be added to these indefinite integrals.5. Change of VariablesTo use integration tables correctly, you must be able to change variables. For example:In xLun xLdxLnduILndu udx xxxILnn LnFHIK zzzsinsinsin sin220221422 20LLet Then use 6. Integration By Partsudv uv vdu zzD. Acosta Page 2 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes RefresherApproximationsFor small x, the following expansions are useful:1.1112 xx x 2. Binomial Expansion:1 1 x nxna f3. Taylor Expansion:f x f xdfdxx d fdxxxaf af 0202 220!Differential EquationsWe will study the solutions to several differential equations when we study circuits in thisclass. Although you are not required to have taken a course in differential equations, we will learn how to solve the simplest ones:1. The exponential function is the only function whose derivative is the function itself:dfdxf xf x Ce Cx is the general solution, where and are constants ( )( )2. Two derivatives of the trigonometric functions give you back the same function with a sign change:d fdxk f xf x A kx B kx A B k222 ( )( ) sin cos , , is the general solution, where and are constantsUsing complex exponentials (see next section), you can also represent this solution as:f x A e B ei k x i k xaf D. Acosta Page 3 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes RefresherComplex NumbersThe analysis of circuits in this class can be greatly simplified by the usage of complex numbers.Complex (or imaginary) numbers are based on i 1 . A complex number may be represented by z x iy , where x and y are real numbers. It can be represented by a point on a two-dimensional plane:An alternative way to represent a complex number is z re r ii cos sinafTo find the magnitude of a complex number (the length r), multiply the number by its complex conjugate, then take the square root:| || |**z z z re re rz z z x iy x iy x yi i or, afaf2 2Note that to take the complex conjugate, replace i with –iIt is possible to represent the sine and cosine functions by complex exponentials:cossin e ee eii ii i22D. Acosta Page 4 1/14/2019 x iy z = x + iy r PHY2061 Enriched Physics 2 Lecture Notes RefresherVectors (see also Appendix H in HRK, vol.2)- Dot product: projection of one vector along anothercosx x y y z za b a b a bab q�= �= + +� =a b b aa b- Cross product: product of vectors, with direction given by right-hand rule( )( )( )detsinx y z y z y z x z x z x y x yx y za a a a b b a a b b a a b b ab b bab q� =- � = = - - - + -� =i j ka b b a i j ka bVector CalculusThe gradient is the rate of change of a function along each direction: ( )ˆ ˆˆx y zx y zgrad f f� � ��= + +� � �= =�FIt yields a vector function when applied to a scalar function.The Laplacian operator is:2 2 222 2 2x y z� � �Ѻ���=++� � �When applied to a scalar function, it yields a scalar result.The line integral is the integral of a vector function projected along a one-dimensional path:( )bC add t dtdt� = �� �� �� �rF r F rThe curve C is the path of function r(t), where t is a variable that parameterizes the path.The line integral is the inverse of the gradient:22 11( ) ( )PPf P f P f d- = � ��rThis means that for conservative forces f=�F, it doesn’t matter what path one takes to go from P1 to P2, the line integral is just the function evaluated at the endpoints. Also, if the path is a closed loop, the line integral equals zero.D. Acosta Page 5 1/14/2019P1P2PHY2061 Enriched Physics 2 Lecture Notes RefresherThe surface integral is the integral of a vector function projected onto a two-dimensionalsurface:( ), S Sd dd u v du dvdu dv� �״=� � �� �� �� �� �r rF s F rThe surface S is parameterized by two variables, u and v.The divergence of a vector function:( )yxzFFFdivx y z����� = = + +� � �F FThe divergence is the flux out of a volume V, per unit volume, as 0V �:01limSVdV��-��F F A�Divergence Theorem: V SdV d�� = �� �F FΑ�The curl of a vector function:( )y yx xz zF FF FF Fcurly z x z x y� �� � � �� �� �� �Ѵ = = - - - + -� � � �� �� � � � � �� �� � � �F F i j kThe curl (when projected along the normal to a loop) is the circulation of a vector function around a closed loop of area A, per unit area, as 0A �:01ˆ( ) limCAdA�Ѵ-��F n F s�Stokes’ Theorem:( )S Cd dѴ�=�� �F A F s�D.
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