hutchings2

hutchings2 - Multivariable Calculus Lecture 5 University of...

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Unformatted text preview: Multivariable Calculus Lecture 5 University of California, Berkeley September 7 1 Brief Review We can describe points in the plane by the xy coordinates. But we can also use the polar coordinates which work better in some situations: 1 Brief Review We can describe points in the plane by the xy coordinates. But we can also use the polar coordinates which work better in some situations: x = r cos , y = r sin . 1 Brief Review We can describe points in the plane by the xy coordinates. But we can also use the polar coordinates which work better in some situations: x = r cos , y = r sin . Given an expression 1 Brief Review We can describe points in the plane by the xy coordinates. But we can also use the polar coordinates which work better in some situations: x = r cos , y = r sin . Given an expression r = f ( ) 1 Brief Review We can describe points in the plane by the xy coordinates. But we can also use the polar coordinates which work better in some situations: x = r cos , y = r sin . Given an expression r = f ( ) we obtained a parametrized planar curve 1 Brief Review We can describe points in the plane by the xy coordinates. But we can also use the polar coordinates which work better in some situations: x = r cos , y = r sin . Given an expression r = f ( ) we obtained a parametrized planar curve x = f ( )cos , y = f ( )sin . Example Example r = 1 + sin . Example r = 1 + sin .-1-0.5 0.5 1 0.5 1 1.5 2 And another example similar to what you saw already last time: And another example similar to what you saw already last time: r = sin( e )- 2 And another example similar to what you saw already last time: r = sin( e )- 2 If the curve is given by r = f ( ) , a b , then the area swept by the position vector moving along the curve is If the curve is given by r = f ( ) , a b , then the area swept by the position vector moving along the curve is Area = 1 2 Z b a f ( ) 2 d . Example: Sketch the curve given by r = 1 + (2 / 3)sin and find the area enclosed by it. Example: Sketch the curve given by r = 1 + (2 / 3)sin and find the area enclosed by it. Answer: Example: Sketch the curve given by r = 1 + (2 / 3)sin and find the area enclosed by it. Answer:-1-0.5 0.5 1 0.5 1 1.5 Example: Sketch the curve given by r = 1 + (2 / 3)sin and find the area enclosed by it. Answer:-1-0.5 0.5 1 0.5 1 1.5 Area = 11 9 2 Points in three space We are now moving to three dimensions where points are described by three coordinates, 2 Points in three space We are now moving to three dimensions where points are described by three coordinates, P = P ( x,y,z ) . Distance: Distance: If P 1 = P 1 ( x 1 ,y 1 ,z 1 ) and P 2 = P 2 ( x 2 ,y 2 ,z 2 ) are two points then the distance between them is Distance: If P 1 = P 1 ( x 1 ,y 1 ,z 1 ) and P 2 = P 2 ( x 2 ,y 2 ,z 2 ) are two points then the distance between them is | P 1 P 2 | = p ( x 1- x 2 ) 2 + ( y 1- y 2 ) 2 + ( z 1- z 2 ) 2 ....
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hutchings2 - Multivariable Calculus Lecture 5 University of...

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