Review of Coordinate Systems, and Vectors

Vector and Scalar Fields

• The field concept is related to a certain region and it is defined at every point in the region.

• A field, scalar or vector, may be defined mathematically as some function of that vector which connects an arbitrary origin to a general point in space.

x

z1 1 1( , , )P x y z

1x

1y

1z

r

1 1 xx x a=

1 1 1x y zr x a y a z a= + +2 2 21 1 1yr x y z= + +

1 1 1

2 2 21 1 1

x y zr

y

x a y a z arar x y z

+ += =

+ +ρ

1 1x yx a y aρ = +

1 1 yy y a=

1 1 zz z a=

y

The Cartesian coordinate system.

A pair of vectors A and B shown in (a) are added by the head-to-tail method (b) and by completing the trapezoid (c). In (d), the vector B is subtracted from A.

Example

The rectangular coordinate system. (a) The axes of the coordinate system and the unit vectors. (b) Location of a point as the intersection of three constant-coordinate planes.

The differential surfaces in a rectangular coordinate system.

Illustration of components of a force vector in moving an object.

Multiplication of Vectors

1. Dot Product

2. Cross Product

The dot product of two vectors.

The cross product of two vectors and the right-hand rule for determining the

direction of the resultant.

Cylindrical Coordinate System

Cylindrical coordinate system.

Shows conversion of the point P(3, 4, 5) in Cartesian coordinates to its equivalent point in cylindrical coordinates.

The cylindrical coordinate system illustrating the unit vectors and the location of a point as the intersection of

three constant-coordinate surfaces.

A differential element in cylindrical coordinates.

Spherical Coordinates System

z

φ

θ

r

x

y

x

y

z

aθ

),,( φθrP

o30

φ

The spherical coordinate system is represented by the orthogonal points (r, θ, Φ).

The spherical coordinate system illustrating the unit vectors and the location of a point as the

intersection of three constant-coordinate surfaces.

A differential element in the spherical coordinate system. One of the element's six surfaces is shaded with the differential surface vector indicated.

Illustration of differential arc lengths in a spherical coordinate system.

(a) Differential arc length for a constant φ. (b) Differential arc length for a constant φ.

The differential surfaces in a spherical coordinate system.

More Examples of spherical surfaces and shapes

θ = 30o θ = 60o

θ = 90o

Spherical Polar Coordinates

θ = 30o θ = 60o

θ = 90o

θ

ϕ

Spherical Polar Coordinates

Illustration of differential arc lengths in a spherical coordinate system.

(a) Differential arc length for a constant φ. (b) Differential arc length for a constant φ.

The differential surfaces in a spherical coordinate system.

A differential element in the spherical coordinate system. One of the element's six surfaces is shaded with the differential surface vector indicated.

Illustration of the line integral; determination of the component of a vector along the path.

Illustration of the surface integral; determination of the component of a vector perpendicular to the

surface.

Figure 2-18 (p. 43)Example 2.8.

Example

Example 2.10.