Multivariable Calculus MATH 2110
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This 2 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2110 at University of Connecticut taught by Iddo Ben-Ari in Fall. Since its upload, it has received 6 views. For similar materials see /class/205805/math-2110-university-of-connecticut in Mathematics (M) at University of Connecticut.
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Date Created: 09/17/15
FORMULA SHEET Vectors 0 Length of vector a 190 llta bcgtl V02 b2 02 o Dot product 017027033 1311921233 albl agbg agbg laHblcosQ7 0 g 9 g 7139 angle radians between vectors 0 Cross product i j k a17a27a3gt X 51527550 a1 a2 a3 7 b1 b2 3 cross product is perpendicular to both vectors on right hand side and its direction is deter mined by right hand rule Also llta17a27aggt x b17b27b3gtl laHbl sin0 same 9 as above 0 Distance from point 17 34121 to a plane ax by 02 d 0 lam 5241 021 dl D V02 b2 02 Flat Geometry 0 Equation of plane with normal vector 11 0 am by 02 d O 0 Equation of line with direction vector 11 c passing through mo7 yo 20 96950 7 yiyo 7 2 20 a 7 b 7 c 7 If any of the coef cients 11 c is equal to 07 then the corresponding quotient is removed from the equation7 and we have x mo when a 07 y yo when I 0 and z 20 when 0 0 Space Curves 0 Unit tangent vector to parametrized curve rt Tt Unit normal vector Nt m WW 0 Arc length of part of curve rt measured between t Oz and t B is lr tldt o Curvature of parametrized curve Mt l 273 W where the rst equation is when the curve is parametrized by arc length 3 Functions of several variables 0 Directional derivative of f in the direction of the unit vector u Duf Vf u 0 Linear approximationtangent plane to function m y at 07 yo 2 mo yoz07 y0z7 3 0 y mwo y 7 yo Change of variables 0 Polar coordinates x rcos0y rsin 0 ffD fzydA ff r cos 07rsin 0rdrd0 the integration on the right hand side is over the appropriate region in the T0 plane 0 Cylindrical coordinates r02 z r cos 0 y r sin 0 2 fffomyzdA fff rcos rsin zrdrd0dz the integration on the right hand side is over the appropriate region in the T02 space 0 Spherical coordinates x osin gtc0s07 y psin gt sin 02 pcos j fffD fzyzdV f f f fp sin 1 cos 0 p sin 1 sin 0 p cos 102 sin gtdpd gtdt97 the integration on the right hand side is over the appropriate region in the p0 gt space Line Integrals C smooth curve parametrized by rt7 04 g t g B 0 f0 fo ls7 d3 lrtldt 0 f0 F dr fF Tds where T is the tangent vector de ned in the section on space curves above f0 de Qdy Rdz fCF dr where F P QR o Green s theorem If C encloses a planar domain D7 then fCltP7 Q dr ffD 7 dA7 when C has positive orientation essentially counterclockwise Surface Integrals S smooth curve parametrized by ruv7 where 7471 are in some planar domain D o ru x rwuo7 U0 is a normal vector to the tangent plane through ruo7 v0 Us de HD mm mm x ruiltwgtdudu o If S is oriented with normal 71 and F is a vector eld7 then ffSF 15 ffSF ndS
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