COORDINATE SYSTEMS & TRANSFORMATION 1.6 CARTESIAN COORDINATES 1.7 CIRCULAR CYLINDRICAL COORDINATES 1.8 SPHERICAL COORDINATES 1.9 TRANSFORMATION BETWEEN COORDINATE SYSTEMS

1.6 CARTESIAN COORDINATES Consists of three mutually orthogonal axes and a point in space is denoted as

CARTESIAN COORDINATES (Cont’d) The location of the point is defined by the intersection of three planes. Example of location of points and

CARTESIAN COORDINATES (Cont’d) Unit vector of in the direction of increasing coordinate value.

CARTESIAN COORDINATES (Cont’d) From previous point P, increase each coordinate value by a differential amount and obtain three slightly displaced planes intersecting at point P’, whose coordinates are , and

CARTESIAN COORDINATES (Cont’d) The six planes define a rectangular parallelepiped whose: Volume: Surface with differential area: Increment in length:

CARTESIAN COORDINATES (Cont’d) Differential surfaces:

1.7 CIRCULAR CYLINDRICAL COORDINATES Form by three surfaces or planes: Plane of z (constant value of z) Cylinder centered on the z axis with radius of Plane perpendicular to x-y plane and rotate about the z axis by angle of Unit vector of in the direction of increasing coordinate value.

CIRCULAR CYLINDRICAL COORDINATES (Cont’d)

CIRCULAR CYLINDRICAL COORDINATES (Cont’d) By increasing each coordinate value with a differential increment , it produces a shape of truncated wedge.

CIRCULAR CYLINDRICAL COORDINATES (Cont’d) The truncated wedge whose: Volume: Surface with differential area: Increment in length:

CIRCULAR CYLINDRICAL COORDINATES (Cont’d) Remember ?! Increment in length for direction: is not increment in length!

CIRCULAR CYLINDRICAL COORDINATES (Cont’d) Differential surfaces:

CIRCULAR CYLINDRICAL COORDINATES (Cont’d)

CIRCULAR CYLINDRICAL COORDINATES (Cont’d) The variables of Cartesian and cylindrical coordinate systems are easily related to each other.

EXAMPLE 5 A cylinder with radius of and length of . Determine: The volume enclosed. The surface area of that volume.

SOLUTION TO EXAMPLE 5 (i) For volume enclosed, we integrate:

SOLUTION TO EXAMPLE 5 (Cont’d) (ii) For surface area of the volume:

EXAMPLE 6 The surfaces define a closed surface. Find: The enclosed volume. The total area of the enclosing surface. The length of the longest straight line that lies entirely within the volume.

SOLUTION TO EXAMPLE 6 (a) The enclosed volume: The limits on the integration must be converted to radians!!

SOLUTION TO EXAMPLE 6 (Cont’d) (b) The total area of the enclosed surface:

SOLUTION TO EXAMPLE 6 (Cont’d) (c) Performing these points to Cartesian coordinates, these become: Taking A and B as vectors directed from the origin, the requested length is:

1.8 SPHERICAL COORDINATES Point P in spherical coordinate can be denoted as .  distance from origin  angle between the z axis and the line from origin to point P  angle between x axis and projection in z=0 plane

SPHERICAL COORDINATES (Cont’d) By increasing each coordinate value with a differential increment , a differential volume can be constructed.

SPHERICAL COORDINATES (Cont’d)

SPHERICAL COORDINATES (Cont’d) However, the increment of length is different from the differential increment previously, where:  distance between two radius  distance between two angles  distance between two radial planes at angles

SPHERICAL COORDINATES (Cont’d) It has: Volume: Surface with differential area: Increment in length:

SPHERICAL COORDINATES (Cont’d) Differential surfaces:

SPHERICAL COORDINATES (Cont’d)

SPHERICAL COORDINATES (Cont’d) The variables of Cartesian and spherical coordinate systems are easily related to each other.

EXAMPLE 7 Find the area of the strip which is a section of sphere with 3cm

SOLUTION TO EXAMPLE 7 Use differential surface with constant radius R, which is in direction.

Differential in length VERY VERY IMPORTANT!! Cartesian Cylindrical Spherical Differential in length Differential Surface Differential Volume

1.9 TRANSFORMATION Transformation of Cartesian, cylindrical and spherical coordinate are obtain geometrically. (i) Transformation between Cartesian and cylindrical:

TRANSFORMATION (Cont’d) (ii) Transformation between Cartesian and spherical:

EXAMPLE 8 Let (Circular cylindrical vector) be converted to (Cartesian vector).

SOLUTION TO EXAMPLE 8 For Cartesian vector: Let Cartesian vector component, So,

SOLUTION TO EXAMPLE 8 (Cont’d) From figure, WHY ?! So,

SOLUTION TO EXAMPLE 8 (Cont’d) For: And,

SOLUTION TO EXAMPLE 8 (Cont’d) Finally, Vector for Cartesian coordinate Put into matrix form:

IMPORTANT!! From previous example, we can conclude that the transformation emphasizing on the dot product, where:

EXAMPLE 9 Given in Cartesian coordinates, Evaluate: Point P in Cylindrical coordinates Vector at P in Cartesian coordinates system Vector in Cylindrical coordinates system Vector at P in Cylindrical coordinates system

SOLUTION TO EXAMPLE 9 (i) In Cartesian coordinates, point P Transformation for point P:

SOLUTION TO EXAMPLE 9 (Cont’d) (ii) Vector at P in Cartesian coordinates systems is easily obtained as: (iii) Vector in Cylindrical coordinates systems

SOLUTION TO EXAMPLE 9 (Cont’d) Where,

SOLUTION TO EXAMPLE 9 (Cont’d) But So, Vector in Cylindrical coordinates systems

SOLUTION TO EXAMPLE 9 (Cont’d) (iv) For vector at P in Cylindrical coordinates system, plug in the values of coordinate P in Cylindrical coordinate into vector below:

SOLUTION TO EXAMPLE 9 (Cont’d) Therefore,