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

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Unformatted text preview: 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 x (x ,y ) x y x y z z z x y (x ,y ,z ) 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 x z x y (x ,y ,z ) z Right Handed z (x ,y ,z ) 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 L e f t H a n d e d R ig h t H a n d e d 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....
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w1-C - Math 20C Multivariable Calculus Lecture 1 1 Slide 1...

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