Divergence Calculator

A vector calculus calculator computes three fundamental differential operators on vector and scalar fields: divergence (∇·F), curl (∇×F), and the Laplacian (∇²f). These operators appear throughout physics and engineering — in Maxwell's equations for electromagnetism, the Navier–Stokes equations for fluid flow, the heat equation, and the wave equation. This tool evaluates them numerically at a given point by accepting user-supplied partial derivative values.

Divergence measures the net outward flux of a vector field per unit volume at a point (source/sink strength). Curl measures the rotational tendency of a vector field (circulation per unit area). The Laplacian of a scalar field f equals the divergence of the gradient: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²; it appears in Laplace's equation (∇²f = 0) and Poisson's equation (∇²f = ρ).

1. Divergence tab: enter the three partial derivatives ∂P/∂x, ∂Q/∂y, ∂R/∂z of your vector field F = (P, Q, R) evaluated at the point of interest. The calculator computes div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
2. Curl tab: enter the six cross-partial derivatives. The calculator computes each component of curl F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y) and the magnitude |curl F|.
3. Laplacian tab: enter the three second-order partial derivatives ∂²f/∂x², ∂²f/∂y², ∂²f/∂z² of a scalar field f. The calculator returns ∇²f.

Vector calculus operator formulas

Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z. Gauss's divergence theorem: ∯_S F·dA = ∭_V (∇·F) dV. For incompressible fluid flow: ∇·v = 0.

Curl: ∇×F = (∂R/∂y − ∂Q/∂z) î + (∂P/∂z − ∂R/∂x) ĵ + (∂Q/∂x − ∂P/∂y) k̂. Stokes' theorem: ∮_C F·dr = ∬_S (∇×F)·dA. Conservative fields: ∇×F = 0.

Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z². Laplace equation: ∇²f = 0 (harmonic). Poisson equation: ∇²φ = −ρ/ε₀ (electrostatics). Heat equation: ∂T/∂t = α∇²T.

Interpreting vector calculus results

Physical interpretation of the operators

Divergence > 0 at a point means that point is a source of the field (e.g. positive charge in electrostatics). Divergence < 0 means a sink. Divergence = 0 everywhere means the field is solenoidal (no sources or sinks — true for magnetic fields: ∇·B = 0). Curl = 0 everywhere means the field is irrotational and conservative — a potential function exists (e.g. gravitational and electrostatic fields). A non-zero curl indicates rotation, as in the magnetic field around a current-carrying wire (Ampère's law: ∇×B = μ₀J).

Math tips and best practices

• This tool accepts numerically evaluated partial derivatives — you must first differentiate your field symbolically and then substitute the coordinates of the point of interest before entering values.
• For a 2D vector field F = (P(x,y), Q(x,y)), the only non-zero component of curl F is the z-component: ∂Q/∂x − ∂P/∂y (the scalar curl or vorticity).
• The Laplacian of a harmonic function is zero everywhere in its domain — such functions are smooth with no local maxima or minima in the interior (maximum principle).
• In cylindrical or spherical coordinates, the operator expressions are different — the Cartesian formulas used here are only valid in Cartesian coordinates.

Divergence, curl, and Laplacian formulas are standard results from vector analysis and are expressed in Cartesian coordinates. This calculator is for educational and numerical evaluation purposes only and does not perform symbolic differentiation. For symbolic computation, use a computer algebra system such as Mathematica, Maple, or SymPy.

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