18 02 Final Exam Formula Sheet Exam 1 Dot product r r r r A B A B cos r r A B a1b1 a2b2 Cross product r r r r A B A B sin n Area of parallelogram is equal to half the length of the prod Matrix arithmetic 1 Addition A B add corresponding elements must be same shape 2 Multiplication AB C multiply each element of row i by column j then add 3 Inversion A 1 A M minors C cofactors J adjoint divide by A 1 M just determinant of the minors C checkerboard signs J flip around top left bottom right diagonal One or many solutions If A is nonzero the matrix has exactly one solution If A 0 the matrix has either 0 or solutions Lines in parametric r With P0 x0 y0 z0 and v a b c x at x0 y bt y0 z ct z0 Planes in parametric r With P0 x0 y0 z0 and v a b c a x x0 b y y0 c z z0 0 r r because N v 0 Given P1 P2 P3 Find P1 P2 P1 P3 r Take prod to get N r Given surface ax by cz N a i b j c k More parametric dx dy dz i v t j k dt dt dt 18 02 Final Exam Formula Sheet Intersection of curve surface sub curve eqn into surface Angle between two planes Find N1 N 2 Take prod Product r r rule forr and r r dB r dA d A B A B dt dt dt r r r r d A B r dB r dA A B dt dt dt Exam 2 Tangent plane z z z z0 x x0 y y0 x y f x0 y0 z0 x x0 y y0 z z0 f f f x x0 y y0 z z0 x y z Normal vector z z i j k if z f x y x y f x0 y0 z0 Max min Find partials set to zero critical points d saddle d fxxfyy fxy2 d fxx min fxx max Chain rule z f u v z f du f dv x u dx v dx u u x v v x Gradient r f f f i j x y magnitude steepness always points uphill 18 02 Final Exam Formula Sheet Directional derivative r r R dz f r z s ds R Lagrange multipliers f x y restraint g x y L x y f x y x y L L L take set equal to zero solve system x y Exam 3 Double integrals b d a c f x y dydx Do inner as usual treat outer variable as constant Do outer Polar coordinates f r rdrd r Integration applications Mass R dA Center of mass x Average of f over R R x dA mass Moment of inertia R area R Work line integrals r r F dr Mdx Ndy C C Paramaterize x and y as functions of t Gradient fields For a gradient field C t1 t0 M dx dy N dt dt dt r r F dr f end f start M N Gradient field test y x To find the potential function integrate Mdx and Ndy Green s theorem f x y dA distaxis 2 dA 18 02 C Final Exam Formula Sheet N M dA R x y r r F dr area Mdx Ndy If Nx My 1 C Flux C M N dA R x y Ndx Mdy Exam 4 Triple integrals D f x y z b a g2 x g1 x h2 x y h1 x y f x y z dzdydx get outer and middle variables from shadow Cylindrical coordinates f r z rdzdrd r z Spherical coordinates f 2 sin d d d Applications of 3D integrals Mass D dV y r cos x r sin z z x sin cos y sin sin z cos Average of f over D D f x y z dV volume D Moment of inertia D distance from axis 2 dV Center of mass D x dV mass Gravitational attraction G D cos sin dV Surface integrals Find S F n ds 1 Inspection F n is constant S F n ds F n area s 2 Cylindrical spherical coords z F n radius dzd F n radius 2 sin d d 3 Rectangular case S is z f x y ncyl x i y j radius nsph x i y j z k radius 18 02 Final Exam Formula Sheet S F n ds shadow F fxi fyj k dA Divergence theorem S is closed surface oriented outward F ds D div F dV div F Mx Ny Pz F Flux Stokes theorem F dr S F ds Work Properties of div curl grad 1 div curl F 0 F 0 2 curl gradient field 0 gradient field test F 0
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