Scalar and Vector Point Functions

Scalar Quantity

A scalar quantity is described as a quantity that has only magnitude but no particular direction. For example – volume, energy, speed, mass, density, and time.

Vector Quantity

Vector quantity is described as a quantity that has both magnitude and direction. For example – force and velocity.

Scalar and Vector Point of Functions

Scalar point functions

A scalar point function is defined as a function which assigns  a real number to every point of a part of the region of space.

If to every point (p, q, r) or z of a region x in space, there is assigned a real number m = Φ(p, q, r), then Φ is called a scalar point function.

Vector Point Functions

A vector point function is defined as a function which assigns  a vector to every point of a part of the region of space.

If to every point (p, q, r ) or z of a region x in space, there is assigned a vector V = V(p, q, r), the V is called a vector point function and the function is represented as:

V = v₁ (p, q, r) i + v₂ (p, q, r) j +v₃ (p, q, r) k

OR,

V = v₁      v₁ (p, q, r)

      v₂  =  v₂ (p, q, r)

      v₃       v₃ (p, q, r)

Examples of Scalar Point Function and Vector Point Function

Examples of Scalar Point Functions are:

1. If z = (p, q, r) then p₂ + q₂ + r₂ is a scalar point function and it forms a three dimensional scalar field.
2. If z = (p,q) then (z) = p₂+q₂ is a scalar point function and it forms a two dimensional scalar field.

Examples of Vector Point Function are:

1. ∇ = pi + qj + zk is a vector point function, which associates with each point (p, q, r)) a vector pointing away from its origin. This represents a three-dimensional source field.

Scalar Function and Vector Function

Scalar Functions

To set of real variables A scalar Function is that which assigns a real number. It is represented in general form as,

m = m(P₁, P₂, P₃, …, P_n )

Where, P₁, P₂, P₃, …, P_n) can be considered real numbers.

Vector Functions

A vector function is that which assigns a group of real variables to the vector. It is represented in general form as,

V = v₁ (P₁ , P₂, P₃, … , P_n) i + v₂ (P₁ , P₂, P₃ … , P_n) j + v₃ (P₁, P₂, P₃ … , P_n) k

Where, P₁, P₂, P₃, …., P_n can be considered real numbers.

Scalar Field and Vector Field

Scalar field

A Scalar field is a scalar point function that is defined over some region. A scalar field that is not dependent on time is called a  steady-state scalar field or stationary.

A scalar field varying with time is represented as,

m = Φ (p, q, r, t)

Examples of scalar fields are –

• The temperature at each point in an insulated wall
• The mass density of the atmosphere
• The water pressure at each point in an ocean

Vector Field

A Vector field is a vector point function defined over some region. A vector field that is not dependent on time is called a steady-state vector field or stationery.

A vector field varying with time is represented as,

V = v₁ (p, q, r, t) i + v₂ (p, q, r, t) j + v₃ (p, q, r, t)k

Examples of vector fields defined in space are –

• Magnetic Field
• Gravitational Field

Conclusion

This is to conclude that a scalar point function is defined as a function which assigns a real number to every point of a part of the region of space and a vector point function is defined as a function which assigns a vector to every point of a part of the region of space.