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by: Nikki Bee

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MA 261 Midterm 2 Study Guide MATH 261

Marketplace > Purdue University > Mathematics (M) > MATH 261 > MA 261 Midterm 2 Study Guide
Nikki Bee

GPA 4.0
Multivariate Calculus
Patricia E Bauman

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COURSE
Multivariate Calculus
PROF.
Patricia E Bauman
TYPE
Study Guide
PAGES
3
WORDS
KARMA
50 ?

Popular in Mathematics (M)

This 3 page Study Guide was uploaded by Nikki Bee on Saturday October 10, 2015. The Study Guide belongs to MATH 261 at Purdue University taught by Patricia E Bauman in Fall 2015. Since its upload, it has received 35 views. For similar materials see Multivariate Calculus in Mathematics (M) at Purdue University.

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Date Created: 10/10/15
MA 261 Spring 2011 Study Guide 2 3 3 f f properties of gradients gradient points in 81 8y direction of maximum rate of increase of f The maximum value of the directional derivative is equal to Vfa0 yo J level curve fa y C and7 in the case of3 variables Vfa0 yo 20 I level surface f1 y z C Gradient vector for fay Vf337y z Y0 20 n Vfxo Yo x x fxYC lt x0 5y0 2 O fXYrZC Relativelocal extrema critical points V f 6 or V f does not exist 2nd Derivatives Test A critical points is a local min if D fmfyy 3y gt 0 and fm gt 07 local max if D gt 0 and fm lt 07 saddle if D lt 0 absolute extrema Max Min Problems Lagrange Multipliers Extremize f subject to a constraint 92 2 C solve the system V f AVg and 92 2 C m n Double integrals Midpoint Rule for rectangle f f 3334 dA Z Z ay i AA R 21 31 h1y S 33 S h2y 9132 33139200 Type I region D Type II region D C S y S d 7 agxgb 92 11 b iterated integrals over Type I and II regions f 13 y dA f 13 y dy d3 and D 1 9130 d h2y f 51334 dA f 3334 d3 dy respectively Reversing Order of Integration regions D c h1y that are both Type I and Type II properties of double integrals Integral inequalities mA g dA 3 MA Where A 2 area of D and m S g M D on D Change of Variables Formula in Polar Coordinates if D hlw S T S h26 then ozlt6 6 h209 fxydA f7 cos67 rsin6rdrd6 D a 1110 T 6 Applications of double integrals a Area of region D is AD dA D b Volume of solid under graph of z f1 y Where fa y 2 0 is V fa y dA D c Mass of D is m p1y dA Where p1 y 2 density per unit area sometimes write D m dm Where dm 2 p1 y dA D d Moment about the az aXis M9 y p1 y dA moment about the y aXis My 2 1 p1 y dA D D MyDap33ydA Mm f ypx7ydA e Center of mass 5 y Where E 7 g E pct y M D m fDpw7ydf1 Remark centroid 2 center of mass When density is constant this is useful 7 Elementary solids E C R3 of Type 1 Type 2 Type 3 triple integrals over solids E fayzdv fxyzdz dA for E 33731 6 D May S 2 3 W731 VE volume of solid E is dV applications of triple integrals mass of a solid moments E about the coordinate planes Mmy Mm Myz center of mass of a solid 3amp2 8 Cylindrical Coordinates 7quot 6 2 Z xIYIz 1 7 cos6 From CC to RC y rsin6 z z AX Going from RC to CC use 32 y2 r2 and tan6 y make sure 6 is in correct quadrant 1 9 Spherical Coordinates p 6 gb Where 0 3 gb 3 7r 2 xIYIz 1 psingb cos6 From SC to RC y psin gb sin6 z pcosgb pcos s V Going from RC to SC use 32 y2 Z2 2 p2 tan6 g and cosgb E a p 10 11 12 1 rcos6 y rsin6 22 ffE xvvadVffEfrcos6rsin6zrdzdrd6 T de39rdzd39rdH Triple integrals in Cylindrical Coordinates 1 psin gb cos6 y psingb sin6 z pcosgb ffEfxyz dVEfpsingbcos6 psingbsinQ pcos p2sjn dpd d9 T dV 2 p2 singb dp dqb d0 Triple integrals in Spherical Coordinates Vector elds 011 R2 and R3 x 3 135 3 62013731 and FOE y Z 135 3 62613731 R013 31 F is a conservative vector eld if F V f for some real valued function f

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