Math 20C Multivariable CalculusLecture 11Slide 1’&$%Coordinates in space•Overview of vector calculus.•Coordinate systems in space.•Distance formula. (Sec. 12.1)Slide 2’&$%Vector calculus studies derivatives and integralsof functions of more than one variableMath 20A studies:f:R→R,f(x), differential calculus.Math 20B studies:f:R→R,f(x), integral calculus.Math 20C considers:f:R2→R,f(x, y);f:R3→R,f(x, y, z);r:R→R3,r(t) =hx(t), y(t), z(t)i.
Math 20C Multivariable CalculusLecture 12Slide 3’&$%Incorporate one more axis toR2and one getsR3Every point in a plane can be labeled by an ordered pairof numbers, (x, y).(Descartes’ idea.)Every point in the space can be labeled by an orderedtriple of numbers, (x, y, z).There are two types of coordinates systems in space asidefrom rotations:Right handedandLeft handed.The same happens inR2.Slide 4’&$%Every point in the plane can be labeled by anordered pair of numbers,(x, y)xyy0x0(x ,y )000xy0x0y0z00zzxy(x ,y ,z )000Every point in the space can be labeled by anordered triple of numbers,(x, y, z)
Math 20C Multivariable CalculusLecture 13Slide 5’&$%There are two types of coordinate systems exceptby rotationsy0x0z0xy(x ,y ,z )000zRight Handed00z0(x ,y ,z )000zLeft HandedyxyxNone rotation transforms a one into the otherSlide 6’&$%This coordinate system is right handedzyxxzyzxyThis coordinate system is left handedzzxyxyzxy
Math 20C Multivariable CalculusLecture 14Slide 7’&$%The same happens inR2xxyyLeft HandedRight HandedInR3we will define thecross productof vectors.The definition gives different results in left handed or inright handed coordinate systems.(There is no cross product inR2.)We use right handed coordinate systemsSlide 8’&$%The distance between points in space is crucial togeneralize the idea of limit to functions in spaceTheorem 1The distance between the pointsP1= (x1, y1, z1)andP2= (x2, y