# calculus1 - The following is a list of formulae and...

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The following is a list of formulae and theorems of math 23 to help you to memorizethe materials and thus no proofs or examples are included. If a formula holds for both2D and 3D cases and the expressions are similar, we only state the 3D case.(1) Distance formula: given two pointsP= (x1, x2, x3), Q= (y1, y2, y3)R3, thenthe Euclidean distance ofpandQisd(P, Q) =|PQ|=(x1-y1)2+ (x2-y2)2+ (x3-y3)2.(2) Equation of spheres and description of balls:The equation of the sphere with center (a, b, c) and radiusris(x-a)2+ (y-b)2+ (z-c)2=r2orx2+y2+z2-2ax-2by-2cz+a2+b2+c2-r2= 0.Please see the later classification of quadric surfaces.The (closed) ball with centerP= (a, b, c) and radiusrisB(P, r) =Br(P) ={(x, y, z)R3|(x-a)2+ (y-b)2+ (z-c)2r2}.(3) Properties of vectors: given vectorsv=x, y, zandw=a, b, cwe haveThe length ofvis|v|=x2+y2+z2.v±w=x±a, y±b, z±c.Ifαis a scalar, thenαv=αx, αy, αz.vis the zero vector if and only ifx=y=z= 0.Ifv= 0, then the unit vector in the direction ofvis1|v|v.The standard basis vectorsi,j,khave componentsi=1,0,0 ,j=0,1,0 ,k=0,0,1 . We can writevas a linear combinationv=xi+yj+zk.(4) Products of vectors: given vectorsv=x, y, zandw=a, b, c, the dot productofvandwis the scalarv·w=xa+yb+zc.Properties of dot product:• |v|2=v·v.v·w=w·v.For scalarsα, βand vectorsv1, v2, wwe have(αv1+βv2)·w=αv1·w+βv2·w.Letθbe the angle betweenvandw. Thenv·w=|v||w|cosθ.For nonzero vectorsvandw, they are perpendicular if and only ifv·w= 0.The scalar projection ofwontoviscompvw=v·w|v|and the vector ofwontovisprojvw=v·w|v|v|v|=v·w|v|2v.1
The cross product of two 3D vectorsv=x, y, zandw=a, b, cis the vectorv×w= deti,j,kx,y,za,b,c= (yc-bz)i+ (za-xc)j+ (xb-za)k.Properties of dot product:v×w=-w×v.For scalarsα, βand vectorsv1, v2, wwe have(αv1+βv2)×w=αv1×w+βv2×w.Letθbe the angle betweenvandw. Then|v×w|=|v||w|sinθ.The vectorv×wis perpendicular to bothvandwand the triple (v, w, v×w)satisfy the right hand rule. Furthermore,|v×w|is the area of the parallelo-gram spanned byvandw.For nonzero vectorsvandw, they are parallel if and only ifv×w= 0.For vectorsu, v, wwe haveu·(v×w) = (u×v)·wandu×(v×w) = (u·w)v-(u·v)w.The volume of the parallelepiped spanned byu, v, wisV=|u·(v×w)|.(5) Lines: a lineLinR3is determined by a pointQ0= (x0, y0, z0) onLand a nonzerovectorv=a, b, cwhich is parallel toL. The vector equation isr=r0+tvwherer0=--→OQ0andtis a parameter. The parametric equation isx=x0+aty=y0+btz=z0+ctand the symmetric equation isx-x0a=y-y0b=z-z0cwhenabc= 0.

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