GeometryCalculusApproximationsDifferential 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 Refresher Math and Physics Refresher This 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 of some of the concepts used in this course. Please report any inaccuracies to the professor. Geometry 1. Pythagorean Theorem: The square of the hypotenuse of a right triangle is the sum of the squares of the two legs: ca b22=+2 c a b 2. Circumference of a circle: CR=2π 3. Volume of a sphere: VR=433π 4. Surface area of sphere: SR=42π Calculus 1. Differentiation: You are expected to be able to take simple derivatives: ddxxnxddxxxddxxxddxeennxx===−=−1sin coscos sin 2. Product Rule ddxfxgx fxdgdxgxdfdxddxxxxx()() () ()sin cos sin=+=+Example: afx D. Acosta Page 1 8/21/2005PHY2061 Enriched Physics 2 Lecture Notes Refresher 3. Chain Rule ddxfgxdfdgdgdxddxxxddxxxafbgchch==−=− −Examples: sin cosexp exp222222x 4. Integration You are expected to be able to perform simple integrals: xdxnxdx x xdx x xdx e ennxxzzzz=+==−=+111cos sinsin cos A purist would note that a constant should be added to these indefinite integrals. 5. Change of Variables To use integration tables correctly, you must be able to change variables. For example: InxLunxLdxLnduILndu udx xxxILnnLn=FHIK=⇒===−==zzzsinsinsin sin2202214222πππππππ0LLet Then use 6. Integration By Parts udv uv vdu=−zz D. Acosta Page 2 8/21/2005PHY2061 Enriched Physics 2 Lecture Notes Refresher Approximations For small x, the following expansions are useful: 1. 1112−≈+ + +xxx" 2. Binomial Expansion: 11+≈++xnxnaf" 3. Taylor Expansion: fx f xdfdxxdfdxxxafaf≈+ + +==0202220!" Differential Equations We will study the solutions to several differential equations when we study circuits in this class. 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: dfdxfxfx Ce Cx==ααα 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: dfdxkfxfx A kx B kx AB k222=−=+()() sin cos , , is the general solution, where and are constants Using complex exponentials (see next section), you can also represent this solution as: f x Ae Beikx ikxaf=′+′− D. Acosta Page 3 8/21/2005PHY2061 Enriched Physics 2 Lecture Notes Refresher Complex Numbers The 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 zxiy=+, 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 zre r ii==+ θθθcos sinaf xiyz = x + iyrθ To find the magnitude of a complex number (the length r), multiply the number by its complex conjugate, then take the square root: ||||**zzzrererzzz xiyxiy xyii== ===−+=+−θθor, afaf22 Note that to take the complex conjugate, replace i with –i It is possible to represent the sine and cosine functions by complex exponentials: cossinθθθθθθ=+=−−−eeeeiiiii22 D. Acosta Page 4 8/21/2005PHY2061 Enriched Physics 2 Lecture Notes Refresher Vectors (see also Appendix H in HRK, vol.2) • Dot product: projection of one vector along another cosxx yy zzab ab ababθ⋅=⋅= + +⋅=ab baab • Cross product: product of vectors, with direction given by right-hand rule ()()()detsinxyz yzyz xzxz xyxyxyzaa a abba abba abbabbbabθ×=−×= = − − − + −×=ijkab ba ijkab Vector Calculus The gradient is the rate of change of a function along each direction: ()ˆˆˆxyzxyzgrad f f∂∂∇= + +∂∂∂==∇F∂ It yields a vector function when applied to a scalar function. The Laplacian operator is: 2222222xyz∂∂∂∇≡∇⋅∇= + +∂∂∂ 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: ()bCaddtdt⋅= ⋅⎡⎤⎣⎦∫∫rFr Fr dt The 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: 2211() ()PPfPfP fd−=∇⋅∫r This 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. P1 P2 D. Acosta Page 5 8/21/2005PHY2061 Enriched Physics 2 Lecture Notes Refresher The surface integral is the integral of a vector function projected onto a two-dimensional surface: (), SSddduv ddu dv⎛⎞⋅= ⋅ ×⎡⎤⎜⎟⎣⎦⎝⎠∫∫rrFs Fr udv The surface S is parameterized by two variables, u and v. The divergence of a vector function: ()yxzFFFdivxyz∂∂∂∇⋅ = = + +∂∂∂FF The divergence is the flux out of a volume V, per unit volume, as : 0V →01limSVdV→∇⋅ ≡ ⋅∫FFvA Divergence Theorem: VSdV d∇⋅ = ⋅∫∫FFΑv The curl of a vector function: ()yyxxzzFFFFFFcurlyz xz xy∂∂⎛⎞ ⎛∂∂∂∂⎛⎞∇× = = − − − + −⎜⎟ ⎜⎜⎟∂∂ ∂∂ ∂∂⎝⎠⎝⎠ ⎝FF i⎞⎟⎠jk The 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→∇× ⋅ ≡ ⋅∫Fn F sv Stokes’ Theorem: ()SCdd∇× ⋅ = ⋅∫∫FA Fvs D. Acosta Page 6 8/21/2005PHY2061 Enriched Physics 2 Lecture Notes Refresher Physics – Mechanics Newton’s Laws: 1. An object maintains constant velocity unless acted upon by an external force 2. The acceleration of an object
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