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Jamie Little
Part IA Michaelmas, Lent, Easter Term

Scalar and Vector Fields

Definitions

Scalar Field

A scalar field ϕ(r)\phi(\mathbf{r}) assigns a scalar (number) to each point in space.

Examples:

  • Temperature in a room: T(x,y,z)T(x, y, z)
  • Pressure in the atmosphere: P(x,y,z)P(x, y, z)
  • Electric potential: V(r)V(\mathbf{r})

Vector Field

A vector field F(r)\mathbf{F}(\mathbf{r}) assigns a vector to each point in space.

Examples:

  • Velocity of fluid flow: v(x,y,z)\mathbf{v}(x, y, z)
  • Electric field: E(r)\mathbf{E}(\mathbf{r})
  • Gravitational field: g(r)\mathbf{g}(\mathbf{r})

The Gradient Operator

The del or nabla operator ∇\nabla is:

∇=i^∂∂x+j^∂∂y+k^∂∂z\nabla = \hat{\mathbf{i}}\frac{\partial}{\partial x} + \hat{\mathbf{j}}\frac{\partial}{\partial y} + \hat{\mathbf{k}}\frac{\partial}{\partial z}

Gradient of a Scalar Field

∇ϕ=∂ϕ∂xi^+∂ϕ∂yj^+∂ϕ∂zk^\nabla \phi = \frac{\partial \phi}{\partial x}\hat{\mathbf{i}} + \frac{\partial \phi}{\partial y}\hat{\mathbf{j}} + \frac{\partial \phi}{\partial z}\hat{\mathbf{k}}

The gradient is a vector field pointing in the direction of steepest increase of ϕ\phi.

Properties of the Gradient

Direction

∇ϕ\nabla \phi points in the direction of maximum rate of increase of ϕ\phi.

Magnitude

∣∇ϕ∣=rate of steepest increase|\nabla \phi| = \text{rate of steepest increase}

Contour map with gradient vectors pointing uphill, perpendicular to level curves

Directional Derivative

The rate of change of ϕ\phi in direction n^\hat{\mathbf{n}}:

dϕdn=∇ϕ⋅n^\frac{d\phi}{dn} = \nabla \phi \cdot \hat{\mathbf{n}}

This is the component of ∇ϕ\nabla \phi in direction n^\hat{\mathbf{n}}.

Level Surfaces

A level surface of ϕ\phi is the set of points where ϕ\phi is constant:

ϕ(x,y,z)=c\phi(x, y, z) = c

Level surfaces of a scalar field, with gradient vector perpendicular to level surface

Key theorem: ∇ϕ\nabla \phi is perpendicular to the level surface.

Proof: If r(t)\mathbf{r}(t) lies on the surface, then ϕ(r(t))=c\phi(\mathbf{r}(t)) = c. Differentiating:

∇ϕ⋅drdt=0\nabla \phi \cdot \frac{d\mathbf{r}}{dt} = 0

The tangent dr/dtd\mathbf{r}/dt to the surface is perpendicular to ∇ϕ\nabla \phi.

Visualising Fields

Scalar Fields

  • Contour plots: Level curves (2D) or level surfaces (3D)
  • Colour mapping: Assign colours to different values
  • Example: Weather maps with isobars (pressure contours)

Contour plot of a scalar field f(x,y) showing level curves f = constant

Vector Fields

  • Arrow plots: Draw vectors at grid points
  • Streamlines: Curves everywhere tangent to the field
  • Example: Weather maps showing wind velocity

2D vector field visualisation showing arrows at grid points, demonstrating rotation or divergence patterns

The Differential of a Scalar Field

For a small displacement dr=dx i^+dy j^+dz k^d\mathbf{r} = dx\,\hat{\mathbf{i}} + dy\,\hat{\mathbf{j}} + dz\,\hat{\mathbf{k}}:

dϕ=∇ϕ⋅dr=∂ϕ∂xdx+∂ϕ∂ydy+∂ϕ∂zdzd\phi = \nabla \phi \cdot d\mathbf{r} = \frac{\partial \phi}{\partial x}dx + \frac{\partial \phi}{\partial y}dy + \frac{\partial \phi}{\partial z}dz

This is the total differential: the change in ϕ\phi due to a small displacement.

Special Cases

Radial Fields

Distance from origin: r=x2+y2+z2r = \sqrt{x^2 + y^2 + z^2}

∇r=rr=r^\nabla r = \frac{\mathbf{r}}{r} = \hat{\mathbf{r}}

For any function f(r)f(r) of distance:

∇f(r)=f′(r)r^\nabla f(r) = f'(r) \hat{\mathbf{r}}

Example: Gravitational potential ϕ=−GMr\phi = -\frac{GM}{r}

∇ϕ=GMr2r^\nabla \phi = \frac{GM}{r^2}\hat{\mathbf{r}}

The gravitational field (force per unit mass) is g=−∇ϕ\mathbf{g} = -\nabla \phi.

Why Gradient Matters

  • Force fields: F=−∇V\mathbf{F} = -\nabla V (force from potential)
  • Heat flow: q=−k∇T\mathbf{q} = -k\nabla T (heat flows down temperature gradient)
  • Mass transport: J=−D∇C\mathbf{J} = -D\nabla C (diffusion down concentration gradient)

The gradient encodes “downhill” directions for flow and equilibration processes.

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