1. CHAPTER 14 Vectors in three space Team 6: Bhanu Kuncharam Tony Rocha-Valadez Wei Lu

2. The position vector R from the origin of Cartesian coordinate system to the point (x(t), y(t), z(t)) is given by the expression A Cartesian coordinate system (by MIT OCW) 14.6 Non-Cartesian Coordinates The vector expression for velocity is given by The vector expression for acceleration is given by http://www.wepapers.com/Papers/4521/1_Newton's_Laws,_Cartesian_an d_Polar_Coordinates,_Dynamics_of_a_Single_Particle

3. 14.6.1 Plane polar coordinate To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis. Polar Angles: The Polar Angle θ of a point P, P ≠ pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis. The Polar Coordinates (r, θ ) of the point P, P ≠ pole, consist of the distance r of the point P from the Pole and of the Polar Angle θ of the point P. Every (0, θ ) represents the pole. θ Polar Axis r P(r, θ) Definitions: http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html

4. Plane polar coordinate More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are: http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html

5. Plane polar coordinate Difference quotient method to get What is ? Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

6. Plane polar coordinate Difference quotient method to get What is ? Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

7. Transform method to get Plane polar coordinate

8. A polar coordinate system (by MIT OCW) The expressions of R, v, a in polar coordinates http://www.wepapers.com/Papers/4521/1_Newton's_Laws, _Cartesian_and_Polar_Coordinates,_Dynamics_of_a_Sing le_Particle

9.  r  r (r, ,z) Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height z axis. 14.6.2 Cylindrical coordinates A cylindrical coordinate system http://mathworld.wolfram.com/CylindricalCoordinates.html

10. Cylindrical coordinates The relations between cylindrical coordinates and Cartesian coordinates. Definitions: The expressions of position R, velocity v, and acceleration a in Cylindrical coordinates are given by Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

11. Cylindrical coordinates Example1: Find the cylindrical coordinates of the point whose Cartesian coordinates are (1, 2, 3) Answer: Example2: Find the Cartesian coordinates of the point whose cylindrical coordinates are (2, Pi/4, 3) Answer: http://mathworld.wolfram.com/CylindricalCoordinates.html

12. 14.6.3 Spherical coordinates (x,y,z)(x,y,z) r  z  Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. Define to be the azimuthal angle in the -plane from the x-axis with (denoted when referred to as the longitude), to be the polar angle (also known as the zenith angle and colatitude, with where is the latitude) from the positive z-axis with, and to be distance (radius) from a point to the origin.

13. Spherical coordinates The expressions of Spherical coordinates for velocity and acceleration

14. The expressions of R, v, a in Spherical coordinates Figure taken from reference: http://mathworld.wolfram.com/SphericalCoordinates.htmlhttp://mathworld.wolfram.com/SphericalCoordinates.html

15. Example 3 Calculate the three components of the position, velocity and acceleration vectors at t=3. The position of the point R is given by R=(t, exp(t), 3t ). Do this for the in Cartesian coordinates, Cylindrical coordinates, and Spherical coordinates Examples: The expressions of R, v, a in Non-Cartesian coordinates Solution: In Cartesian Coordinates:

16. Solution: In Cylindrical Coordinates: putinto get The expressions of R, v, a in Non-Cartesian coordinates

17. In Spherical Coordinates: put into Solution : get The expressions of R, v, a in Non-Cartesian coordinates

18. Using the omega method derive the space derivatives of base vectors Consider a rigid body B undergoing an arbitrary motion through 3-space. And let A be any fixed vector with B, that is, A is a vector from one material point in B to another so is constant with time, because b is rigid. Thus A=A(t) Fixed vector in B There exists a vector such that Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall. 14.6.4 Omega Method

19. Omega method Since B is arbitrary: Since A is arbitrary: So we get Omega Method

20. Omega method In cylindrical coordinates : Let A be : Using chain differentiation to write: Similarly, let A be : Let A be : Omega Method

21. End of Chapter 14