w1-C - Math 20C Multivariable Calculus Lecture 1 1 $ Coordinates in space Slide 1 Overview of vector calculus Coordinate systems in space Distance

w1-C - Math 20C Multivariable Calculus Lecture 1 1 $...

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Math 20C Multivariable Calculus Lecture 1 1 Slide 1 & $ % Coordinates in space Overview of vector calculus. Coordinate systems in space. Distance formula. (Sec. 12.1) Slide 2 & $ % Vector calculus studies derivatives and integrals of functions of more than one variable Math 20A studies: f : R R , f ( x ), differential calculus. Math 20B studies: f : R R , f ( x ), integral calculus. Math 20C considers: f : R 2 R , f ( x, y ); f : R 3 R , f ( x, y, z ); r : R R 3 , r ( t ) = h x ( t ) , y ( t ) , z ( t ) i .
Math 20C Multivariable Calculus Lecture 1 2 Slide 3 & $ % Incorporate one more axis to R 2 and one gets R 3 Every point in a plane can be labeled by an ordered pair of numbers, ( x, y ). (Descartes’ idea.) Every point in the space can be labeled by an ordered triple of numbers, ( x, y, z ). There are two types of coordinates systems in space aside from rotations: Right handed and Left handed . The same happens in R 2 . Slide 4 & $ % Every point in the plane can be labeled by an ordered pair of numbers, ( x, y ) x y y 0 x 0 (x ,y ) 0 0 0 x y 0 x 0 y 0 z 0 0 z z x y (x ,y ,z ) 0 0 0 Every point in the space can be labeled by an ordered triple of numbers, ( x, y, z )
Math 20C Multivariable Calculus Lecture 1 3 Slide 5 & $ % There are two types of coordinate systems except by rotations y 0 x 0 z 0 x y (x ,y ,z ) 0 0 0 z Right Handed 0 0 z 0 (x ,y ,z ) 0 0 0 z Left Handed y x y x None rotation transforms a one into the other Slide 6 & $ % This coordinate system is right handed z y x x z y z x y This coordinate system is left handed z z x y x y z x y
Math 20C Multivariable Calculus Lecture 1 4 Slide 7 & $ % The same happens in R 2 x x y y Left Handed Right Handed In R 3 we will define the cross product of vectors. The definition gives different results in left handed or in right handed coordinate systems. (There is no cross product in R 2 .) We use right handed coordinate systems Slide 8 & $ % The distance between points in space is crucial to generalize the idea of limit to functions in space Theorem 1 The distance between the points P 1 = ( x 1 , y 1 , z 1 ) and P 2 = ( x 2 , y

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