Scalar and Vector Fields

A scalar field is a function that returns a scalar value for each point in a space. More formally, for set \(P\), \(f: P \to \mathbb{R}\) is a scalar field if for every \(x \in P\), \(f(x)\) is a scalar in \(\mathbb{R}\). In many applications, \(P\) is a Euclidean space; that is, \(P \subset \mathbb{R}^n\) for some positive integer \(n\). For engineering problems, \(n\) is often \(3\) and the vector space is the three-dimensional Euclidean space \(\mathbb{R}^3\) of everyday experience.

Here are some examples of scalar fields:

• For fluid flow, the pressure \(P:\mathbb{R}^3 \to \mathbb{R}\), where \(P(\bm{x})\) is the pressure at point \(\bm{x}\).
• For heat transfer, the temperature \(T: \mathbb{R}^3 \to \mathbb{R}\), where \(T(\bm{x})\) is the temperature at point \(\bm{x}\).
• For continuum mechanics, the density \(\rho: \mathbb{R}^3 \to \mathbb{R}\), where \(\rho(\bm{x})\) is the density at point \(\bm{x}\).
• For electromagnetism, the electric potential \(V: \mathbb{R}^3 \to \mathbb{R}\), where \(V(\bm{x})\) is the electric potential at point \(\bm{x}\).
• For gravity, the gravitational potential \(\Phi: \mathbb{R}^3 \to \mathbb{R}\), where \(\Phi(\bm{x})\) is the gravitational potential at point \(\bm{x}\).

A vector field is a function that returns a vector for each point in a space. More formally, for \(P \subset Q\), \(\bm{F}: P \to Q\) is a vector field if for every \(\bm{x} \in P\), \(\bm{F}(\bm{x})\) is a vector in \(Q\). In many applications, \(P\) and \(Q\) are Euclidean spaces; that is, \(P \subset Q\) and \(Q = \mathbb{R}^n\) for some positive integer \(n\). For engineering problems, as for scalar fields, \(n\) is often \(3\) and the vector space is the three-dimensional Euclidean space \(\mathbb{R}^3\) of everyday experience.

Here are some examples of vector fields that appear in engineering:

• For fluid flow, the velocity field \(\bm{v}: \mathbb{R}^3 \to \mathbb{R}^3\) describes the velocity of the fluid at each point in space.
• For heat transfer, the heat transfer field \(\bm{q}: \mathbb{R}^3 \to \mathbb{R}^3\) describes the heat flow at each point in space.
• For electromagnetic fields, the electric field \(\bm{E}: \mathbb{R}^3 \to \mathbb{R}^3\) and the magnetic field \(\bm{B}: \mathbb{R}^3 \to \mathbb{R}^3\) describe the electric and magnetic forces at each point in space.
• For gravitational fields, the gravitational field \(\bm{g}: \mathbb{R}^3 \to \mathbb{R}^3\) describes the gravitational force at each point in space.
• For mechanical systems in general, the force field \(\bm{F}: \mathbb{R}^3 \to \mathbb{R}^3\) describes the force at each point in space.

In the following sections, we will explore several important operations relating to vector fields, such as the gradient, divergence, curl, line integral, and surface integral. In the fields described above, these operations have important physical interpretations.