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Harold’s Calculus 3Multi-Coordinate SystemCheat Sheet29 November 2022RectangularPolar/CylindricalSphericalParametricVectorMatrixPoint2Dfx= yx, ya, b3Dfx, y=zx, y, z4Dfx, y, z=wx, y, z, w ?(r , θ) or r ∠ θ(ρ , θ, ?) x=ρ sin ? cos θy=ρsin?sinθz=ρ cos ?ρ2= r2+ z2ρ2= x2+y2+ z2tanθ=yx?= cos-1zx2+ y2+ z2?= cos-1zρPoint (a,b) in Rectangular?:xt=ayt=b<a,b>t=3rd variable, usually time,with 1 degree of freedom (df)r= x0, y0, z0r= xi+yj+zkax= bPolar Rect.x=rcosθy=rsinθz=ztanθ=yxRect. Polarr2= x2+y2 r= ± x2+ y2θ=tan-1yxLineSlope-Intercept Form:y=mx+bPoint-Slope Form:y- y0=m (x-x0)General Form:Ax+By+C=0where A and B≠0Calculus Form:fx=f'a x+f(0)where m = f’(a)3-D:x- x0a= y- y0b= z- z0c20129515621000<x,y> = <x0,y0>+ t<a,b><x,y> = <x0+at,y0+bt>where<a,b> = <x2-x1, y2-y1>x(t)= x0+ tay(t)= y0+ tbm=?y?x= y2-y1x2-x1=bar= r0+t v= x0, y0, z0+t a, b, c22542531559500ab xy= cabcd xy= efPlaneax- x0+ by- y0+ cz- z0=0ax+by+cz=dwhere d= ax0+by0+cz0fx, y= Ax+By+C(r , θ, constant) 0≤r< ∞(0≤θ< 2π)where r and θ take on all values in their domains(ρ , θ, constant) 0≤ρ< ∞(0≤θ< 2π)where ρ and θ take on all values in their domainsr= r0+ sv+twwhere:s and t range over all real numbersv and w are given vectors defining the planer0 is the vector representing the position of an arbitrary (but fixed) point on the planen ? r- r0=0ConicsGeneral Equation for All Conics:Ax2+Bxy+Cy2+Dx+Ey+F=0whereLine: A=B=C=0Circle: A=C and B=0Ellipse: AC>0 or B2-4AC<0Parabola: AC=0 or B2-4AC=0Hyperbola: AC<0 or B2-4AC>0Note: If A+C=0, square hyperbolaRotation:If B ≠ 0, then rotate coordinate system:cot2θ=A-CBx=x'cosθ-y'sinθy=y'cosθ+x'sinθNew = (x’, y’), Old = (x, y)rotates through angle θ from x-axisGeneral Equation for All Conics:Vertical Axis of Symmetry:r=p1-ecosθ Horizontal Axis of Symmetry:r=p1-esinθ where p= a 1-e22d a e2-1for 0≤e<1e=1e>1p = semi-latus rectumor the line segment running from the focus to the curve in a direction parallel to the directrixEccentricity:Circle e=0Ellipse 0<e<1Parabola e=1Hyperbola e>1 NACirclex2+y2=r2(x-h)2+(y-k)2=r2General Form:Ax2+Bxy+Cy2+Dx+Ey+F=0where A=C and B=0Center: h,kVertices: NAFocus: h,kCentered at Origin:r = a (constant)θ = θ 0, 2π or [0, 360°]Centered at (r0,?):r2+r02-2rr0cosθ-?=R2Hint: Law of Cosinesorr=r0cosθ-?+a2-r02 sin2(θ-?)ρ= constantθ= θ [0, 2π]?=constant=0x(t)=rcos(t)+hy(t)=rsin(t)+ktmin, tmax=[0, 2π)Center: h,kFocus: h,kNANASpherex2+y2+z2=r2(x-h)2+(y-k)2+(z-l)2=r2 Focus and center:(h, k, l)General Form:Ax2+By2+Cz2+ Dxy+Eyz+Fxz+ Gx+Hy+Iz+J=0where A=B=C > 0Cylindrical to Rectangular:x=r cos?(θ)y=r sin?(θ)z=zSpherical to Rectangular:x=rsinθcos?y=rsinθsin?z=rcosθRectangular to Cylindrical:r= x2+y2Spherical to Cylindrical:ρ=r sin?(θ)?=?z=r cos?(θ)ρ= constantθ= θ [0, 2π]?=? [0, 2π]Rectangular to Spherical:r=x2+y2+z2θ=arccoszr?=arctanyxCylindrical to Spherical:r= ρ2+z2θ= arctanρz=arccoszr?=?Rectangular:r≡xyzCylindrical:r≡r cos?(θ)r sin?(θ)zSpherical:r≡rsinθcos?rsinθsin?rcosθEllipse(x-h)2a2+(y-k)2b2=1General Form:Ax2+Bxy+Cy2+Dx+Ey+F=0where B2-4AC<0 or AC>0Center: h,kVertices: h±a, kCo-Vertices: h, k±bFoci: h±c, kFocus length, c, from center:c2= a2-b2Eccentricity:e= ca= a2- b2aIf B ≠ 0, then rotate coordinate system:cot2θ=A-CBx=x'cosθ-y'sinθy=y'cosθ+x'sinθNew = (x’, y’), Old = (x, y)rotates through angle θ from x-axisVertical Axis of Symmetry:r=a (1-e2)1±ecosθHorizontal Axis of Symmetry: r=a (1-e2)1±esinθEccentricity: 0<e<1rθ=a b(bcosθ)2+(asinθ)2relative to center (h, k)Interesting Note:The sum of the distances from each focus to a point on the curve is constant.d1+d2=kx(t)=acos(t) +hy(t)=bsin(t)+ ktmin, tmax=[0, 2π]Center: h,kRotated Ellipse:xt=acostcosθ-bsintsinθ+hyt=acostsinθ+bsintcosθ+kθ = the angle between the x-axis and the major axis of the ellipseEllipsoid(x-h)2a2+(y-k)2b2+(z-l)2c2=1r2cos2θa2+r2sin2θb2+z2c2=1r2cos2θ sin2?a2+r2sin2θ sin2?b2+r2cos2?c2=1x(t,u)=acos(t) cos(u)+hy(t,u)=bcost sin(u)+ kzt,u=c sin(t)+ l tmin, tmax=-π2, π2umin, umax=[-π, π]Center: h, k,lx-vTA-1x-v=1Centered at vector vHyperbolax-h2a2-y-k2b2=1General Form:Ax2+Bxy+Cy2+Dx+Ey+F=0where B2-4AC>0 or AC<0If A+C=0, square hyperbolaCenter: h,kVertices: h±a, kFoci: h±c, kFocus length, c, from center:c2= a2+b2Eccentricity:e= ca= a2+ b2a=secθIf B ≠ 0, then rotate coordinate system:cot2θ=A-CBx=x'cosθ-y'sinθy=y'cosθ+x'sinθNew = (x’, y’), Old = (x, y)rotates through angle θ from x-axisInteresting Note:The difference between the distances from each focus to a point on the curve is constant.d1-d2=k Vertical Axis of Symmetry:r=a (e2-1)1±ecosθHorizontal Axis of Symmetry: r=a (e2-1)1±esinθEccentricity: e>1where e= ca= a2+ b2a=secθ>1relative to center (h, k)-cos-1-1e<θ<cos-1-1ep = semi-latus rectum or the line segment running from the focus to the curve in the directions θ=± π2Left-Right Opening Hyperbola:x(t)=asec(t)+hy(t)=btan(t)+ktmin, tmax=[-c, c]Vertex: (h, k)Alternate Form:x(t)= ±acosh(t)+ hy(t)=bsinh(t)+ kUp-Down Opening Hyperbola:x(t)=atan(t)+hy(t)=bsec(t)+ktmin, tmax=[-c, c]Vertex: (h, k)Alternate Form:x(t)=asinh(t)+ hy(t)=±bcosh(t)+ kGeneral Form:x(t)=At2+ Bt+Cy(t)=Dt2+ Et+Fwhere A and D have different signsHyperboloidx-h2a2+y-k2b2-z-l2c2=1- x-h2a2 - y-k2b2+z-l2c2=1ParabolaVertical Axis of Symmetry:x2=4 pyx-h2=4p(y-k)Vertex: h,kFocus: h,k+pDirectrix: y=k-pHorizontal Axis of Symmetry:y2=4 pxy-k2=4p(x-h)Vertex: h,kFocus: h+p,kDirectrix: x=h-pGeneral Form:Ax2+Bxy+Cy2+Dx+Ey+F=0where B2-4AC=0or AC=0If B ≠ 0, then rotate coordinate system:cot2θ=A-CBx=x'cosθ-y'sinθy=y'cosθ+x'sinθNew = (x’, y’), Old = (x, y)rotates through angle θ from x-axisVertical Axis of Symmetry:r=ed1±ecosθHorizontal Axis of Symmetry: r=ed1±esinθEccentricity: e=1and d=2pInteresting Note:The distances from a point on the curve to the focus is the same as to the directrix.Vertical Axis of Symmetry:xt= 2pt+ hyt=pt2+k (opens upwards)yt=-pt2-k (opens downwards)tmin, tmax=[-c, c]Vertex: h,kHorizontal Axis of Symmetry:yt= 2pt+ kxt=pt2+h (opens to the right)xt=-pt2-h (opens to the left)tmin, tmax=[-c, c]Vertex: h,kProjectile Motion:xt=x0+vxt+12axt2yt=y0+vyt-16t2 feetyt=y0+vyt-4.9t2 metersvx=vcosθvy=vsinθGeneral Form:x=At2+ Bt+Cy=Lt2+ Mt+Nwhere A and L have the same signParaboloid(x-h)2a2+(y-k)2b2=(z-l)2c2Limitlimx→cf(x)=L1st Derivativef'x=limh→0fx+h-f(x)hf'c=limx→cfx-f(c)x-cf'x=dydx=y'=Dxdydx=dydθdxdθ=drdθsinθ+ rcosθdrdθcosθ-rsinθHint: Use Product Rule fory=rsinθx=r cos θdydx=dydtdxdt , provideddxdt≠0ddtr= r 'Unit tangent vectorTt= r'(t)r'(t) where r't≠02nd Derivativef''x=ddxdydx= d2ydx2=y''=Dxxd2ydx2=ddxdydx=ddθdydxdxdθd2ydx2=ddxdydx=ddtdydxdxdt=ddtdydtdxdtdxdtUnit normal vectorNt= T'(t)T'(t) where T't≠0IntegralFundamental Theorem of Calculus:Fx=abfx dx =Fb-F(a)Riemann Sum:S=i-1nf(yi)(xi-xi-1)Left Sum:S= 1nfa+fa+1n+fa+2n+…+f(b-1n)Middle Sum:S= 1nfa+12n+fa+32n+…+f(b-12n)Right Sum:S= 1nfa+1n+fa+2n+…+f(b)abrt dt= abftdt, abgtdt, abhtdt Double Integralabc(y)d(y)fx,y dx dySame as rectangular, but fx,y?f(ρcos?, ρsin?)Triple Integralabc(z)d(z)e(y,z)g(y,z)fx,y,z dx dy dzSame as rectangular, butfx,y,z?f(ρcos?, ρsin?,z)Same as rectangular, butfx,y,z?f(ρcosθsin?, ρsinθsin?,ρcos?)NANANAInverse FunctionsIf fx= y, then f-1 (y)=xInverse Function Theorem:f-1f'a= 1f'aif y=sin θif y=cosθif y=tanθif y= csc θif y=secθif y=cotθthen θ=sin-1ythen θ=cos-1ythen θ=tan-1ythen θ=csc-1ythen θ=sec-1ythen θ=cot-1yor θ=arcsinyor θ=arccos yor θ=arctanyor θ=arccscyor θ=arcsecyor θ=arccotyNANANAArc LengthL= ab1+[f'x]2 dxProof:?s= x-x02+ y- y02?s = (?x)2+ (?y)2ds= dx2+dy2ds= dx2+dy2dx2dx2ds= dx2+dydx2dx2ds= dx2 1+dydx2ds= 1+dydx2dxL= dsPolar:L= r2+ drdθ2 dθWhere r=f(θ)Circle:L=s=rθProof:L=fraction of circumference?π?(diameter)L=θ2ππ (2r)=rθC = πd = 2πrRectangular 2D:L= αβdxdt2+ dydt2 dtRectangular 3D:L= αβdxdt2+ dydt2+ dzdt2 dtCylindrical:L= t1t2drdt2+r2dθdt2+dzdt2dtSpherical:L= t1t2dρdt2+ρ2sin2φdθdt2+ρ2dφdt2 dtL= abr'(t) dts(t) = 0tr'(u) duNACurvatureκ= y''(1+y'2)32κ(θ)= r2+2r'2-rr''(r2+r'2)32for r(θ)NAκ= z''y'-y''z'2+x''z'-z''x'2+(y''x-x''y')2 (x'2+y'2+z'2)32where f(t) = (x(t), y(t), z(t))κ= d Tdsκ=T'(t)r'(t) κ=r't × r''(t)r'(t)3 (See Wikipedia?: Curvature)PerimeterSquare: P = 4sRectangle: P = 2l + 2wTriangle: P = a + b + cCircle: C = πd = 2πrEllipse: C≈π(a+b)Ellipse: C≈2πa2+b22C≈π 3a+b-(3a+b)(a+3b)C≈π a+b1+3h10+4-3hEllipse: C=4a0π21-k2sin2θ dθh=a-b2a+b2 & k2=1-b2a2NANANAAreaSquare: A = s?Rectangle: A = lwRhombus: A = ? abParallelogram: A = BhTrapezoid: A=B1+ B22 hKite: A = d1 d22Triangle: A = ? BhTriangle: A = ? ab sin(C)Triangle using Heron’s Formula:A= ss-as-bs-cwhere s=a+b+c2Equilateral Triangle: A = ?3s2Frustum: A=13B1+B22hCircle: A = πr?Circular Sector: A = ? r?θEllipse: A = πabA= αβ12[f(θ)]2dθwhere r=f(θ)Proof:Area of a sector:A=s dr= r ?θ dr=12r2 ?θwhere arc length s=r ?θNAA=αβgt f't dtwhere ft=x and gt=yorx(t) = f(t) and y(t) = g(t)Simplified:A=αβyt dx(t)dt dtProof:abfx dxy = f(x) = g(t)dx =dftdtdt=f’(t) dtA=D ?r?u×?r?vdu dvNALateral Surface AreaCylinder: SA = 2πrhCone: SA = πrlSA= 2πabfx 1+[f'x]2 dxFor rotation about the x-axis:SA= 2πy dsFor rotation about the y-axis:SA= 2πx dsds= r2+drdθ2 dθr=fθ, α≤ θ≤βSphere: SA = 4πr?For rotation about the x-axis:SA= 2πy dsFor rotation about the y-axis:SA= 2πx dsds= dxdt2+dydt2 dtif x=ft, y=gt, α≤ t≤βNANATotal Surface AreaCube: SA = 6s?Rectangular Box: SA = 2lw + 2wh + 2hlRegular Tetrahedron: SA = 2bhCylinder: SA = 2πr (r + h)Cone: SA = πr? + πrl = πr (r + l)Sphere: SA = 4πr?Ellipsoid: SA ≈4πapbp+apcp+bpcp31pWhere p ≈1.6075, Relative Error≤1.061%(Knud Thomsen’s Formula)Ellipsoid: S =where Surface of RevolutionFor revolution about the x-axis:A=2π abf(x)1+dydx2 dxFor revolution about the y-axis:A=2π abx1+dxdy2 dyFor revolution about the x-axis:A=2παβrcosθ r2+ drdθ2 dθFor revolution about the y-axis:A=2παβrsinθr2+drdθ2dθSphere: S = 4πr?For revolution about the x-axis:Ax=2πabyt dxdt2+ dydt2 dtFor revolution about the y-axis:Ay=2πabxt dxdt2+ dydt2 dtNANAVolumeCube: V = s?Rectangular Prism: V = lwhCylinder: V = πr?hTriangular Prism: V= BhTetrahedron: V= ? BhPyramid: V = ? BhCone: V = ? Bh = ? πr?hSphere: V=43πr3Ellipsoid: V = 43 πabcfx, y, z dx dy dzfrcosθ, r sin θ, zr dz dr dθfρsin φcosθ,ρsinφ sinθ, ρcosφ… ρ2sinφ dρ dφ dθEllipsoid:V=43πdetA-1Volume of RevolutionDisk MethodV=ab(area of circle) dthicknessRotation about the x-axis:V= abπ fx2dxRotation about the y-axis:V= cdπx2dyWasher MethodRotation about the x-axis:V= abπ {[f(x)]2-[gx]2} dxV=VOuter Disk-VInner DiskShell MethodV=abcircumference hight dxRotation about the y-axis:V=ab2πx fx dxRotation about the x-axis:V=cd2πy gy dyMoments of InertiaI= i=1Nmi ri2=0am r2 drNANAI= V ρr d(r)2 dV(r)(see Wikipedia)Center of MassR=1M i=1Nmi riwhere M= i=1Nmi1D for Discrete:xcm=m1x1+m2x2m1+m22D for Discrete:My=i=1Nmi xi Mx=i=1Nmi yi x=MyM, y=MxM 3D for Discrete:xcm=x=1M i=1Nmi xi ycm=y=1M i=1Nmi yi zcm=z=1M i=1Nmi zi 3D for Continuous:x=1M 0Mx dmy=1M 0My dmz=1M 0Mz dmwhere M=0Mdmand dm=ρ dz dy dxR=1M r dmR=1M V ρr r dVWhere r is distance from the axis of rotation, not origin.Gradient? ?= ?f?xi+?f?yj+?f?zk? ?(ρ,?,z)= ?f?ρeρ+1ρ?f??e?+?f?zez? ?r,θ,?= ?f?rer+1r?f?θeθ+1r sinθ?f??e?? ?x?v= Dvf(x)? ?= ?fi?xjeiejwhere ?=(?1,?2,?3)Line IntegralC f ds= abfrtr'(t) dtNANAC F(r) ?dr= abFrt? r'(t) dtSurface IntegralS f dS=T fxs,t ?x?s × ?x?tds dtWhere xs,t=xs,t, ys,t, zs,tand?x?s × ?x?t= ?(y,z)?(s,t), ?(z,x)?(s,t),?(x,y)?(s,t) NANAS v ?dS=S v?n dS=T vxs,t ??x?s × ?x?tds dt ................
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