Blog

[vc_row][vc_column width=”2/3″][vc_column_text]In order to describe a vector accurately, some specific lengths, directions, angles, projections, or components must be given. There are three simple methods of doing this, and about eight or ten other methods which are useful in very special cases.

We are going to use only the three simple methods, and the simplest of these is the Cartesian or rectangular coordinate system.

Cartesian or rectangular coordinate system

In the Cartesian coordinate system, we set up three coordinate axes mutually at right angles to each other, and call them the x, y, and z axes. It is customary to choose a right-handed coordinate system, in which a rotation (through the smaller angle) of the x axis into the y axis would cause a right-handed screw to progress in the direction of the z axis.  If the right hand is used, then the thumb, forefinger, and middle finger may then be identified, respectively, as the x, y, and z axes.

Fig. 1.a shows a right-handed cartesian coordinate system.

Figure 1.Cartesian coordinate System

A point is located by giving its x, y, and z coordinates. These are, respectively,  the  distances  from  the  origin  to  the  intersection  of  a  perpendicular dropped from the point to the x, y, and z axes.

Fig. 1.b shows the points P and Q whose coordinates are (1, 2, 3) and (2, -2, 1), respectively.

Point P is therefore located at the common point of intersection of the planes x = 1, y = 2, and z = 3, while point Q is located at the intersection of the planes x = 2, y = -2, z = 1.

If we visualize three planes intersecting at the general point P, whose coordinates are x, y, and z, we may increase each coordinate value by a differential amount and obtain three slightly displaced planes intersecting at point P^t , whose coordinates are x + dx, y + dy, and z + dz.

The six planes define a rectangular parallelepiped whose volume is

The surfaces have differential areas dS of dxdy, dydz, and dzdx.

VECTOR COMPONENTS AND UNIT VECTORS

To describe a vector in the Cartesian coordinate system,

• Let us first consider a vector r extending outward from the origin.
• A logical way to identify this vector is by giving the three component vectors, lying along the three coordinate axes, whose vector sum must be the given vector. If the component vectors of the vector r are x, y, and z, then r = x + y + z.
• The component vectors are shown in Fig. 2.a.

Figure 2.Vector Components

• Instead of one vector, we now have three, but this is a step forward, because the three vectors are of a very simple nature; each is always directed along one of the coordinate axes.
• In other words, the component vectors have magnitudes which depend on the given vector (such as r above), but they each have a known and constant direction. This suggests the use of unit vectors having unit magnitude, by definition, and directed along the coordinate axes in the direction of the increasing coordinate values.
• Thus, a_x, a_y, and a_z are the unit vectors in the cartesian coordinate system.
• They are directed along the x, y, and z axes, respectively, as shown in Fig. 2.b.

If the component vector y happens to be two units in magnitude and directed toward increasing values of y, we should then write

A vector r_P pointing from the origin to point P (1, 2, 3) is written

The vector from P to Q may be obtained by applying the rule of vector addition. This rule shows that the vector from the origin to P plus the vector from P to Q is equal to the vector from the origin to Q.

The desired vector from P (1, 2, 3) to Q (2, -2, 1) is therefore

R_PQ = r_Q – r_P = (2 – 1)a_x + (-2 – 2)a_y + (1 – 3)a_z = a_x – 4a_y – 2a_z

The vectors r_P, r_Q, and R_PQ are shown in Fig. 2.c.

Any vector B then may be described by

CIRCULAR CYLINDRICAL COORDINATES

The Cartesian coordinate system is generally the one in which students prefer to work every problem. This often means a lot more work for the student, because many problems possess a type of symmetry which pleads for a more logical treatment.

• The circular cylindrical coordinate system is the three-dimensional version of the polar coordinates of analytic geometry.
• In the two-dimensional polar coordinates, a point was located in a plane by giving its  distance p from the origin, and the angle Φ between the line from the point to the origin  and  an arbitrary radial line, taken as Φ = O

A three-dimensional coordinate system, circular cylindrical coordinates, is obtained by also specifying the distance z of the point from an arbitrary z = O reference plane which is perpendicular to the line ρ = O

Consider any point as the intersection of three mutually perpendicular surfaces.

• These surfaces are a circular cylinder (ρ = constant), a plane (Φ = constant), and another plane (z = constant). This corresponds to the location of a point in a cartesian coordinate system by the intersection of three planes (x = constant, y = constant, and z = constant).
• The three surfaces of circular cylindrical coordinates are shown in Fig. 3.a. Note that three such surfaces may be passed through any point, unless it lies on the z axis, in which case one plane suffices.

Figure 3.Cylindrical Coordinate System

• Three unit vectors must also be defined, but we may no longer direct them along the coordinate axes, for such axes exist only in Cartesian coordinates. Instead, we take a broader view of the unit vectors in cartesian coordinates and realize that they are directed toward increasing coordinate values and are perpendicular to the surface on which that coordinate value is constant (i.e., the unit vector a_x is normal to the plane x = constant and points toward larger values of x).

In a corresponding way we may now define three unit vectors in cylindrical coordinates, a_ρ, a_Φ, and a_z.

The unit vector a_ρ  at a point P(ρ₁, Φ₁, z₁) is directed radially outward, normal  to  the  cylindrical  surface  ρ = ρ₁.  It lies in the planes Φ = Φ₁ and z = z₁. The unit vector a_Φ is normal to the plane Φ = Φ₁, points in the direction of increasing Φ, lies in the plane z = z₁, and is tangent to the cylindrical surface ρ = ρ₁.The unit vector a_z is the same as the unit vector a_z of the cartesian coordinate system.

Fig. 3.b shows the three vectors in cylindrical coordinates. In Cartesian coordinates, the unit vectors are not functions of the coordinates. Two of the unit vectors in cylindrical coordinates, a_ρ and a_Φ, however, do vary with the coordinate Φ, since their directions change. In integration or differentiation with respect to Φ, then, a_ρ and a_Φ must not be treated as constants.

The unit vectors are again mutually perpendicular, for each is normal to one of the three mutually perpendicular surfaces, and we may define a right- handed cylindrical coordinate system as one in which a_ρ x a_Φ = a_z, or (for those who have flexible fingers) as one in which the thumb, forefinger, and middle finger point in the direction of increasing ρ, Φ, and z, respectively.

• A differential volume element in cylindrical coordinates may be obtained by increasing ρ, Φ, and z by the differential increments dρ, dΦ, and dz.
• The two cylinders of radius ρ and ρ + dρ, the two radial planes at angles Φ and Φ + dΦ, and the two horizontal planes at elevations z and z + dz now enclose a small volume, as shown in Fig. 3.c, having the shape of a truncated wedge.
• As the volume element becomes very small, its shape approaches that of a rectangular parallelepiped having sides of length dρ, pdΦ and dz.

Note that dρ and dz are dimensionally lengths, but dΦ is not; ρdΦ is the length.

• The surfaces have areas of ρdρdΦ, dρdz, and ρ dΦdz, and the volume becomes ρdρdΦdz.

Figure 4.Cylindrical Coordinates conversion to Cartesian Coordinates

The variables of the rectangular and cylindrical coordinate systems are easily related to each other. With reference to Fig. 4, we see that

x = ρcos Φ

y = ρsin Φ

z = z

THE SPHERICAL COORDINATE SYSTEM

We have no two-dimensional coordinate system to help us understand the three- dimensional spherical coordinate system, as we have for the circular cylindrical coordinate system. In certain respects we can draw on our knowledge of the latitude-and-longitude system of locating a place on the surface of the earth, but usually we consider only points on the surface and not those below or above ground.

Let us start by building a spherical coordinate system on the three Cartesian axes (Fig. 5.a). We first define the distance from the origin to any point as r.

Figure 5.Spherical Coordinate System

• The surface r = constant is a sphere.
• The second coordinate is an angle e between the z axis and the line drawn from the origin to the point in question.
• The surface e = constant is a cone, and the two surfaces, cone and sphere, are everywhere perpendicular along their intersection, which is a circle of radius r sin e. The coordinate e corresponds to latitude, except that latitude is measured from the equator and e is measured from the North Pole.
• The third coordinate Φ is also an angle and is exactly the same as the angle Φ of cylindrical coordinates. It is the angle between the x axis and the projection in the z = O plane of the line drawn from the origin to the point. It corresponds to the angle of longitude, but the angle Φ increases to the east. The surface Φ = constant is a plane passing through the e = O line (or the z axis).

We should again consider any point as the intersection of three mutually perpendicular surfaces-a sphere, a cone, and a plane-each oriented in the manner described above. The three surfaces are shown in Fig. 5.b.

Three unit vectors may again be defined at any point.

Each unit vector is perpendicular to one of the three mutually perpendicular surfaces and oriented in that direction in which the coordinate increases.

• The unit vector a_r is directed radially outward, normal to the sphere r = constant, and lies in the cone e = constant and the plane Φ = constant.
• The unit vector a_e is normal to the conical surface, lies in the plane, and is tangent to the sphere. It is directed along a line of longitude and points south.
• The third unit vector a_Φ is the same as in cylindrical coordinates, being normal to the plane and tangent to both the cone and sphere. It is directed to the east.

The three unit vectors are shown in Fig. 5.c.

They are, of course, mutually perpendicular,  and  a  right-handed  coordinate  system  is  defined  by  causing a_r x a_e = a_Φ.  Our system is right-handed, as an inspection of Fig.  5.c will show, on application of  the  definition of the  cross  product. The right-hand rule serves to identify the thumb, forefinger, and middle finger with the direction of increasing r, e, and Φ, respectively. (Note that the identification in cylindrical coordinates was with ρ, Φ, and z, and in cartesian coordinates with x, y, and z).

A differential volume element may be constructed in spherical coordinates by increasing r, e, and Φ by dr, de, and dΦ, as shown in Fig. 5.d.

The distance between the two spherical surfaces of radius r and r + dr is dr; the distance between the two cones having generating angles of e and e + de is rde; and the distance between the two radial planes at angles Φ and Φ + dΦ is found to be r sin edΦ, after a few moments of trigonometric thought.

The surfaces have areas of r dr de, r sin e dr dΦ, and r² sin e de dΦ, and the volume is r² sin e dr de dΦ.

References

[1] “Engineering Electromagnetics Sixth Edition” by William H. Hayt, Jr. John A. Buck[/vc_column_text][/vc_column][vc_column width=”1/3″][/vc_column][/vc_row][vc_row][vc_column width=”2/3″][vc_column_text]AUTHORS
1.Bunty B. Bommera
2.Dakshata U. Kamble[/vc_column_text][/vc_column][vc_column width=”1/3″][/vc_column][/vc_row]