Cartesian coordinates in space (Sect. 12.1).

Transcription

1 Cartesian coordinates in space (Sect. 12.1). Overview of Multivariable Calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere. Overview of Multivariable Calculus Mth 132, Calculus I: f : R R, f (), differential calculus. Mth 133, Calculus II: f : R R, f (), integral calculus. Mth 234, Multivariable Calculus: f : R 2 R, f (, ) f : R 3 R, f (,, ) } scalar-valued. r : R R 3, r(t) = (t), (t), (t) } vector-valued. We stud how to differentiate and integrate such functions.

2 The functions of Multivariable Calculus Eample An eample of a scalar-valued function of two variables, T : R 2 R is the temperature T of a plane surface, sa a table. Each point (, ) on the table is associated with a number, its temperature T (, ). An eample of a scalar-valued function of three variables, T : R 3 R is the temperature T of an object, sa a room. Each point (,, ) in the room is associated with a number, its temperature T (,, ). An eample of a vector-valued function of one variable, r : R R 3, is the position function in time of a particle moving in space, sa a fl in a room. Each time t is associated with the position vector r(t) of the fl in the room. Cartesian coordinates in space (Sect. 12.1). Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere.

3 Cartesian coordinates. Cartesian coordinates on R 2 : Ever point on a plane is labeled b an ordered pair (, ) b the rule given in the figure. (, ) Cartesian coordinates in R 3 : Ever point in space is labeled b an ordered triple (,, ) b the rule given in the figure. (,, ) Cartesian coordinates. Eample Sketch the set S = {,, = } R 3. Solution: = > > S

4 Cartesian coordinates. Eample Sketch the set S = { 1, 1 2, = 1} R 3. Solution: S Cartesian coordinates in space (Sect. 12.1). Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere.

5 Right and left handed Cartesian coordinates. Definition A Cartesian coordinate sstem is called right-handed (rh) iff it can be rotated into the coordinate sstem in the figure. (,, ) Right Handed Definition A Cartesian coordinate sstem is called left-handed (lh) iff it can be rotated into the coordinate sstem in the figure. (,, ) Left Handed No rotation transforms a rh into a lh sstem. Right and left handed Cartesian coordinates. Eample This coordinate sstem is right-handed. Eample This coordinate sstem is left handed.

6 Right and left handed Cartesian coordinates Remark: The same classification occurs in R 2 : Right Handed Left Handed This classification is needed in R 3 because: In R 3 we will define the cross product of vectors, and this product has different results in rh or lh Cartesian coordinates. There is no cross product in R 2. In class we use rh Cartesian coordinates. Cartesian coordinates in space (Sect. 12.1). Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere.

7 Distance formula between two points in space. Theorem The distance P1 P 2 between the points P1 = ( 1, 1, 1 ) and P 2 = ( 2, 2, 2 ) is given b P1 P 2 = ( 2 1 ) 2 + ( 2 1 ) 2 + ( 2 1 ) 2. The distance between points in space is crucial to define the idea of limit to functions in space. Proof. Pthagoras Theorem. P 1 P 2 ( ) 2 1 ( ) 2 1 a ( ) 2 1 P 1 P 2 2 = a 2 + ( 2 1 ) 2, a 2 = ( 2 1 ) 2 + ( 2 1 ) 2.

8 Distance formula between two points in space Eample Find the distance between P 1 = (1, 2, 3) and P 2 = (3, 2, 1). Solution: P1 P 2 = (3 1) 2 + (2 2) 2 + (1 3) 2 P1 P 2 = = 8 We conclude that P1 P 2 = 2 2. Distance formula between two points in space Eample Use the distance formula to determine whether three points in space are collinear. Solution: d 21 P 2 d 32 P 3 P 2 d 32 P 3 d 21 P 1 d 31 P 1 d 31 d 21 + d 32 > d 31 Not collinear, d 21 + d 32 = d 31 Collinear.

9 Cartesian coordinates in space (12.1) Overview of vector calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space. Equation of a sphere. A sphere is a set of points at fied distance from a center. Definition A sphere centered at P = (,, ) of radius R is the set S = { P = (,, ) : P P } = R. R Remark: The point (,, ) belongs to the sphere S iff holds ( ) 2 + ( ) 2 + ( ) 2 = R 2. ( iff means if and onl if. )

10 An open ball is a set of points contained in a sphere. Definition An open ball centered at P = (,, ) of radius R is the set B = { P = (,, ) : P P } < R. A closed ball centered at P = (,, ) of radius R is the set B = { P = (,, ) : P P } R. Remark: The point (,, ) belongs to the open ball B iff holds ( ) 2 + ( ) 2 + ( ) 2 < R 2. Equation of a sphere Eample Plot a sphere centered at P = (,, ) of radius R >. Solution: R

11 Equation of a sphere Eample Graph the sphere =. Solution: Complete the square. = 2 + = [ ( 4 ( 4 2 ] ( 4 ) ) 2) 2 = 2 + ( + 4 2) = 2 + ( + 2) = 2 2. Equation of a sphere Eample Graph the sphere =. Solution: = 2 + ( + 2) = 2 2. Then, we conclude that P = (, 2, ) and R = 2. Therefore, 2

12 Equation of a sphere Eample Given constants a, b, c, and d R, show that a 2b 2c = d is the equation of a sphere iff holds d > (a 2 + b 2 + c 2 ). (1) Furthermore, show that if Eq. (1) is satisfied, then the epressions for the center P and the radius R of the sphere are given b P = (a, b, c), R = d + (a 2 + b 2 + c 2 ).