Scalar Field & Its Gradient

Concept of a Field

By a field we mean any physical quantity that can be associated with every point of a region in space. The quantity associated with each point may be a scalar or a vector. A familiar example of a scalar field is the temperature field inside a room: every point of the room can be assigned a temperature value which, in general, varies from place to place. A familiar example of a vector field is the gravitational field or the electric field, where at each point in space we associate a vector that has both magnitude and direction.

Scalar field versus Vector field

A scalar field gives a single number (a scalar) for every point of the region. Examples: temperature, electric potential, pressure. A vector field gives a vector for every point. Examples: gravitational acceleration, electric field, magnetic field.

Graphical representation

A scalar field is commonly represented by a grid or by contour lines (curves joining points of equal scalar value), for example isotherms on a temperature map. A vector field is represented by drawing arrows (vectors) at chosen points; the arrows show the local direction and magnitude of the field.

In two dimensions we typically use an x-y plane and draw sample vectors at grid points. A convenient choice of scale (unit length for vector arrows) keeps the diagram readable.

Figure 1: Graphical representation of a vector field

In electrostatics, the field due to charges is often shown as lines of force (field lines). The direction of the field at a point is tangent to the field line through that point; the density of the lines indicates field strength. For example, the field lines around a dipole and around two like charges are standard illustrations.

The figure above shows the field due to two opposite charges; a small positive test charge placed at a point experiences a force along the tangent to the field line in the direction shown by the arrow.

Directional Derivatives

We first recall the derivative of a function of a single variable. For f(x), the derivative at x gives the instantaneous rate of change and can be interpreted as the slope of the curve at that point. In higher dimensions we have partial derivatives: for a function f(x,y), the partial derivative with respect to x is the rate of change when y is held fixed, and similarly for y.

When both or all independent variables are allowed to change simultaneously we need the concept of the directional derivative. The directional derivative gives the rate of change of a scalar function when we move from a point P0 in a specified direction by a small distance Δs.

Let φ(x,y,z) be a scalar function. Starting from a point P0, if we move a small distance Δs along some direction to reach point P, the change in φ is given by the directional derivative along that direction times Δs. The directional derivative depends on the direction chosen; there are infinitely many directions in space, so the derivative is a function of direction as well as position.

Using coordinates, the total change in φ for small displacements Δx, Δy, Δz can be expressed in terms of partial derivatives.

For small changes we write the first-order change in φ as the sum of partial derivatives times the corresponding coordinate changes:

Using partial derivatives with respect to x, y and z we obtain the directional derivative along the chosen direction.

Directional derivative formula (vector form)

If a unit vector in the chosen direction is u = (a, b, c), then the directional derivative of φ at a point in that direction is

D_u φ = ∂φ/∂x · a + ∂φ/∂y · b + ∂φ/∂z · c.

Equivalently, if we define the gradient operator ∇ by ∇ = (∂/∂x, ∂/∂y, ∂/∂z) then

D_u φ = ∇φ · u.

Worked example (directional derivative)

Example : We illustrate directional derivative calculations by using a two-variable scalar function and computing derivatives along specified directions at a given point. The function used in the illustration below is shown by the placeholder.

The calculation steps follow the standard method: compute partial derivatives, form the gradient vector, choose the unit direction vector for the given direction, and take the dot product of gradient with that unit vector to obtain the directional derivative. The example in the original text evaluates directional derivatives at the point (1,2) for three directions; the function has a cup-like surface when plotted (a radially symmetric bowl produces circular level curves).

For a two-variable function φ(x,y), the partial derivatives are

To compute the directional derivative along a direction where coordinates satisfy a relation between x and y (for example y = α x) one can parameterise the direction, obtain the unit displacement vector, and use the dot product with the gradient to find the derivative along that direction. The intermediate relations and substitutions are shown in the following placeholders.

Plugging these relations into the expression for the directional derivative yields the directional derivative value at the chosen point. In the worked example the directional derivative at the point (1,2) was evaluated for three directions and, depending on the direction, took different values; one of the directions gave maximum directional derivative, another gave zero.

The directional derivative is maximum when the direction aligns with the gradient vector. In the example the maximum occurs when a parameter (called α in the worked text) takes a value α = 2, showing that the direction of maximum increase is the radial direction at that point.

Gradient of a Scalar Field

The gradient of a scalar field φ(x,y,z) is the vector field formed by its partial derivatives. By definition,

∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z).

The gradient is itself a vector field: at each point it gives the direction and rate of maximum increase of the scalar function. If θ is the angle between ∇φ and a unit direction u, then

D_u φ = ∇φ · u = |∇φ| cos θ.

From this relation the following two important properties follow:

• the magnitude |∇φ| is the maximum possible value of the directional derivative at that point, and
• the direction of ∇φ is the direction in which the directional derivative takes this maximum value.

Physically, this is intuitive: if φ represents height on a hill, the gradient at a point points in the direction of steepest ascent and its magnitude equals the steepness (slope) in that direction. If one wants to descend fastest, one should move in the direction opposite to the gradient.

Equipotential / Level surfaces and normality

A level surface (or equipotential surface in physics) of a scalar function φ is the set of points where φ has the same constant value. Level curves in two dimensions are special cases. The gradient is always normal (perpendicular) to the level surface at a point on that surface. This follows directly from the directional derivative formula because tangent displacements on the level surface produce zero change of φ.

For example, for a radially symmetric function such as a bowl-shaped surface, level curves are concentric circles and the gradient points radially outward, normal to these circles.

In the case illustrated, the gradient takes a radial form and is perpendicular to the circular level curves, which confirms the general property that the gradient is normal to level surfaces.

Formal proof that ∇φ is normal to level surfaces

Consider a level curve (or surface) defined by φ(x,y) = constant. Parameterise a point on this level curve by a parameter t (for the circle, t may be the polar angle θ). Let r(t) be the position vector of a point on the curve. Then the tangent vector to the curve is dr/dt.

On a level curve φ(r(t)) is constant for all t. Differentiating with respect to t gives

∇φ · (dr/dt) = 0.

This shows that the gradient ∇φ is orthogonal to dr/dt, the tangent to the level curve. Hence the gradient is normal to the level surface at that point.

Summary and applications

The key points to remember are:

• Scalar field: a number assigned to every point of a region (examples: temperature, potential).
• Vector field: a vector assigned to every point of a region (examples: electric and magnetic fields, gravitational acceleration).
• Directional derivative: the rate of change of a scalar field when moving from a point in a specified direction; for a unit direction u it equals ∇φ · u.
• Gradient: ∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z); it is a vector field that points in the direction of maximum increase of φ and whose magnitude is that maximum rate of increase.
• Level surfaces: surfaces of constant φ; the gradient at any point is perpendicular to the level surface through that point.

These concepts are fundamental in electrostatics and other areas of physics and engineering. For example, electric field E in electrostatics is related to the electric potential V by E = -∇V, so the field lines are orthogonal to equipotential surfaces and point in the direction of greatest decrease of potential.