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HARVARD MATH 21A - Checklists

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1 Checklist Week 1 I Definitions coordinates vectors sphere dot product cross product triple cross product parallel vectors orthogonal vectors scalar projection Compv w and vector projection Projv w II Facts v w v1 w1 v2 w2 v3 w3 v w cos v w v w sin is area of paralellogram u v w volume of parallelepiped III Algorithms Adding subtracting and scaling vectors geometrically as well as algebraically Completion of square Compute dot cross triple products Find distance between points Find vector orthogonal to two vectors Area of parallelogram volume of a parallelepiped spanned by three vectors 2 Checklist Week 2 Line plane ax by cz d x x0 t v sw Domain and range of functions f x y Graph G x y f x y Intercepts intersections of G with coordinate axes Traces intersections with coordinate planes Generalized traces intersections with x c y c or z c Quadric Ellipsoid Paraboloid One and Two sheeted Hyperboloid Cylinder Cone Hyperbolic Paraboloid I Definitions 0 0 y y z z b c x x0 t v x x0 a II Facts Plane ax by cz d has normal vector n a b c 0 0 0 y y z z contains v a b c Sets g x y z c describe Line x x a b c surfaces Special case graphs g x y z z f x y Linear equation i e 2x 3y 5z 7 defines planes Quadratic equation i e x2 2y 2 3z 2 4 defines quadric III Algorithms Using dot and cross product to derive distance formulas distance point plane distance point line line line Geometric constructions example finding plane through P Q R Intersection of two planes intersection of line plane angles between lines and placnes switch from different descriptions of lines and planes Sketch and match graphs of f x y Sketch and match quadrics Completion of squares to find type of quadric 1 3 Checklist Week 3 I Definitions Plane and space curves r t Velocity r 0 t Acceleration r 00 t Rt Position from velocity r t 0 r0 s ds r 0 Unit tangent T t r 0 t r 0 t t T 0 t T 0 t Unit normal N t Binormal vector B t T t N 0 0 Curvature t T t r t Rb Arc length a r 0 t dt II Facts r 0 t is tangent to the curve Rt v r 0 then r 0 v dt c r 0 t r 00 t r 0 t 3 d Identities like dt v t t w t v 0 t w t v t w 0 t T N B are unit vectors which are perpendicular to each other III Algorithms Compute r 0 t T t for curve r t Draw curves in the plane or in space Match curves with their parametric equations Find parameterizations of curves i e intersections of surfaces 4 Checklist Week 4 I Definitions polar x y r cos r sin cylindrical x y z r cos r sin z spherical x y z cos sin sin sin cos g r 0 polar curve especially r f polar graphs g r z 0 cylindrical surface especially r f z or r f z surface of revolution g 0 spherical surface especially f r u v x u v y u v z u v parametrized surface Fix one variable grid curves II Examples x2 y 2 r2 r t r cos t r sin t x2 y 2 z 2 2 r cos sin sin sin cos ax by cz d r u v r0 u v v w a b c v w r z g z r u v g v cos u g v sin u v g x y z z f x y 0 r u v u v f u v III Algorithms plot curves and surfaces from implicit or parametric equation match curves and surfaces with equations parametrize curves and surfaces parametric description r u v implicit description g x y z 0 for planes sphere graphs surfaces of revolution translate from and into polar coordinates translate from and into spherical or cylindrical coordinates 2 5 Checklist Week 5 I Definitions fx x y x f x y partial derivative partial differential equation PDE like F f fx ft fxx ftt 0 The function f is the unknown ft fxx heat equation ftt fxx 0 wave equation ih ft fxx 0 Schro dinger fx ft 0 transport equation fx f ft 0 Burgers equation fxx fyy 0 Laplace equation L x y f x0 y0 fx x0 y0 x x0 fy x0 y0 y y0 linear approximation Tangent line L x y L x0 y0 or ax by d with a fx x0 y0 b fy x0 y0 d ax0 by0 Tangent plane L x y z L x0 y0 z0 or Can estimate f x y y by L x y z near x0 y0 z0 f x y differentiable that is if fx fy continuous II Facts fxy fyx Clairot s theorem ru u v rv are tangent to surface III Algorithms Distinguish ODE s and PDE s Verify that given function satisfies a PDE Compute tangent lines and tangent planes Estimate f x0 dx y0 dy for small dx dy Decide about differentiability of f x y 6 Checklist Week 6 I Definitions f x y fx fy f x y z fx fy fz gradient spelled Nabla D v f f v directional derivative we define it like this for all v L x f x0 f x0 x x0 linearization II Facts d r t dt f f r t r 0 t chain rule f x0 orthogonal to level set f x c containing x0 d x dt f t v Dv f x by chain rule x x0 f x 0 tangent space at x0 Directional derivative maximal in v f direction Partial derivatives are special directional derivatives If Dv f x 0 for all v then f x 0 III Algorithms Implicit differentiation example f x y x 0 compute y 0 x fx fy without knowing y x use chain rule fx 1 fy y 0 x 0 Compute directional derivatives D v f Find direction where directional derivative is maximal Find tangent lines and tangent planes Estimate f x y near a point f x 0 y0 as L x y where L x y is the linearization of f 3 7 Checklist Week 7 I Definitions f x y 0 0 critical point or stationary point 2 D fxx fyy fxy discriminant or Hessian determinant f x0 y0 f x y in a neighborhood of x0 y0 local maximum f x0 y0 f x y in a neighborhood of x0 y0 local minimum f x0 y0 f x y for all x y global maximum or absolute maximum f x0 y0 f x y for all x y global minimum or absolute minimum f x y g x y g x y c Lagrange multiplier Two constraints f g h g c h d II …


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HARVARD MATH 21A - Checklists

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