PHYS2150 Chap.6 Gradient, Divergence and Curl

Gradient, Divergence and Curl

Begin by identifying the field type

A scalar field assigns one number to every point, such as temperature, pressure, density or potential. A vector field assigns a vector, such as velocity, gravitational acceleration or electric field. The del operator combines with these fields in different ways. Gradient accepts a scalar and returns a vector. Divergence accepts a vector and returns a scalar.

Curl accepts a three-dimensional vector field and returns another vector. Checking input and output types before differentiating prevents many notation errors.

Gradient gives steepest increase and surface normals

For a differentiable scalar field φ, grad φ=∇φ is built from its partial derivatives. The directional derivative in a unit direction u is Duφ=∇φ·u.

Its largest possible value is ||∇φ|| and occurs in the gradient direction; the smallest occurs in the opposite direction. On a level surface f(x,y,z)=k, the gradient is normal to the surface, so it supplies the normal vector for a tangent plane and the direction of a normal line.

Divergence measures local source strength

For F=(P,Q,R), div F=∇·F=P_x+Q_y+R_z.

In a fluid-velocity model it represents net outflow per unit volume near a point. Positive divergence resembles a source, negative divergence a sink, and zero divergence means locally balanced inflow and outflow. A divergence-free field can still be nonzero and can still rotate.

Applying divergence to a gradient gives the Laplacian, ∇²φ=φ_xx+φ_yy+φ_zz, which reappears in heat, wave and Laplace equations.

Curl measures local rotational tendency

Curl is the cross product ∇×F written through component derivatives. Its direction follows the right-hand rule and its magnitude measures local twisting. A field with zero curl is irrotational.

For rigid-body rotation, curl of the velocity field equals twice the angular-velocity vector. Curl and divergence describe independent features: a circulating field may have zero divergence and nonzero curl, while another field can have nonzero divergence and zero curl.

Potentials are established constructively

If F=∇U, then F is conservative and U is a scalar potential.

The course proves that every sufficiently smooth gradient field has zero curl, so nonzero curl rules out a potential. In the zero-curl examples treated here, construct U directly: integrate one component, retain an unknown function of untouched variables, compare the remaining components, and verify by differentiating the completed potential.

This method supplies the requested scalar and checks every sign rather than stopping at an irrotationality test.

Vector identities provide checks

Important identities include curl grad φ=0, div curl F=0, product rules for scalar–vector products, and curl curl F=grad div F−∇²F. Their smoothness assumptions matter because mixed partial derivatives must agree.

These identities can expose an algebra error or prove that a proposed field cannot be a curl, but they do not replace a requested component calculation.

State the identity and conditions before using it as a shortcut.

Curvilinear coordinates include scale factors

Cylindrical and spherical unit vectors vary with position, so their gradient, divergence and curl formulas contain factors absent in Cartesian coordinates. In cylindrical coordinates, for example, divergence begins with (1/r)∂(rF_r)/∂r and the angular derivative carries 1/r.

Spherical formulas include r² and sinθ factors. First write the field in the correct coordinate basis, use the course angle convention, and retain every scale factor. A Cartesian conversion can verify simple examples but may be longer for genuinely symmetric fields.

What this chapter covers

• 01

  Scalar and vector fields

• 02

  Gradient and level surfaces

• 03

  Divergence and sources

• 04

  Curl and rotation

• 05

  Conservative fields and potentials

• 06

  Laplacian and del identities

• 07

  Cylindrical and spherical operators

Worked example · free

Recover a field potential

Key terms

Conservative Field

A vector field equal to the gradient of a scalar potential on its domain.

Irrotational Field

A vector field whose curl vanishes throughout the region considered.

Directional Derivative

The scalar rate ∇φ·u at which a scalar field changes in a specified unit direction.

Divergence

A scalar measuring local net outward tendency of a vector field.

Curl

A vector whose direction and magnitude describe local rotational tendency.

Laplacian

The scalar operator ∇² obtained by taking divergence of a gradient.

Potential Function

A scalar field U whose gradient equals the given conservative vector field.

Incompressible Field

A vector field with zero divergence throughout the region considered.

Gradient, Divergence and Curl FAQ

Exam move

Translate the wording before doing algebra: steepest increase or a level-surface normal means gradient; source, sink or incompressible means divergence; rotation, vorticity or irrotational means curl; potential means component integration plus a domain check. Write the expected answer type—scalar or vector—in the margin.

Practise one Cartesian field and one cylindrical or spherical field side by side so the scale factors become visible. For potential problems, always retain a function of untouched variables and finish with a full gradient verification. Build a compact identity sheet, but include smoothness assumptions and never use an identity to skip a computation the question explicitly requests.

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