Divergence & Curl Calculator

Overview: Divergence and Curl of a Vector Field

This tool computes the divergence and curl of a three-dimensional vector field at a specific point. In vector calculus and multivariable calculus, these two differential operators help you understand how a field expands, compresses, and rotates in space. They are central in physics, engineering, and applied mathematics, especially in areas like fluid dynamics and electromagnetism.

You can think of a vector field F(x, y, z) as assigning a vector to every point in 3D space, for example a velocity vector of a fluid, or the electric or magnetic field in Maxwell’s equations. The calculator assumes a Cartesian coordinate system and a field of the form:

F(x, y, z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z))

Formulas for Divergence and Curl

The divergence and curl are defined using partial derivatives of the components of F with respect to x, y, and z.

Divergence ∇·F

The divergence measures the net outflow of the vector field from an infinitesimal volume around a point. In Cartesian coordinates:

∇·F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

Curl ∇×F

The curl measures the local rotation or swirling of the field. In Cartesian coordinates the curl is another vector field whose components are:

∇×F = ( ∂F_z/∂y − ∂F_y/∂z,  ∂F_x/∂z − ∂F_z/∂x,  ∂F_y/∂x − ∂F_x/∂y )

Physically, the direction of the curl vector indicates the axis of rotation, and its magnitude indicates how strong that local rotation is.

How to Use the Divergence & Curl Calculator

1. Enter the vector field components.
   • In F_x(x, y, z), type the x-component, e.g. x*y, y^2 + sin(z), or 3*x.
   • In F_y(x, y, z), type the y-component, e.g. y*z or cos(x).
   • In F_z(x, y, z), type the z-component, e.g. z*x or x^2 + y^2.

   Use x, y, and z as the variables. Supported syntax includes standard operators (+, −, *, /, ^) and common functions like sin, cos, exp, and log.

2. Specify the evaluation point.
   • x₀: the x-coordinate of the point, e.g. 1 or 0.5.
   • y₀: the y-coordinate.
   • z₀: the z-coordinate.

   The calculator evaluates the divergence and curl at this single point in space.

3. Click “Evaluate”.

   The tool differentiates each component symbolically with respect to x, y, and z, then substitutes the point (x₀, y₀, z₀) to give numerical values for ∇·F and ∇×F.

Interpreting the Results

The output will show:

• Divergence ∇·F(x₀, y₀, z₀): a single scalar value.
• Curl ∇×F(x₀, y₀, z₀): a vector with three components.

Typical interpretations in vector calculus and physics are:

• Positive divergence: the point behaves like a source (more field is flowing out than in), often associated with fluid being created or charge density being positive.
• Negative divergence: the point behaves like a sink (more field is flowing in than out).
• Zero divergence: locally incompressible or solenoidal behavior; volume elements are neither expanding nor contracting.
• Large curl magnitude: strong local rotation or vorticity, for example a swirling fluid or a rotating electric field.
• Zero curl: locally irrotational; in simply connected regions this is typical of conservative force fields such as gravitational or electrostatic fields in electrostatics.

Worked Example

Consider the vector field

F(x, y, z) = (x y,  y z,  z x)

Step 1: Compute the divergence:

• ∂F_x/∂x = ∂(x y)/∂x = y
• ∂F_y/∂y = ∂(y z)/∂y = z
• ∂F_z/∂z = ∂(z x)/∂z = x

So

∇·F = y + z + x

Step 2: Compute the curl components:

• (∇×F)_x = ∂F_z/∂y − ∂F_y/∂z = 0 − y = −y
• (∇×F)_y = ∂F_x/∂z − ∂F_z/∂x = 0 − z = −z
• (∇×F)_z = ∂F_y/∂x − ∂F_x/∂y = 0 − x = −x

So

∇×F = ( −y,  −z,  −x )

Step 3: Evaluate at the point (1, 2, 3):

• ∇·F(1, 2, 3) = 2 + 3 + 1 = 6
• ∇×F(1, 2, 3) = ( −2,  −3,  −1 )

Entering fx = x*y, fy = y*z, fz = z*x and x0 = 1, y0 = 2, z0 = 3 into the calculator will reproduce these results.

Comparison: Divergence vs Curl

Assumptions and Limitations

• Coordinate system: The calculator works in 3D Cartesian coordinates only, with variables x, y, and z.
• Smoothness: The formulas assume that F_x, F_y, and F_z are differentiable near the evaluation point. Non-differentiable or piecewise-defined fields may give misleading results.
• Symbolic differentiation: The underlying engine uses symbolic rules. Extremely complicated expressions, undefined operations (such as division by zero at the point), or non-standard syntax can cause errors.
• Single-point evaluation: This tool evaluates divergence and curl at a single point, not over a whole region. For full field visualization or plots, you would need additional software.
• Physical modeling: Interpreting the numerical values in a real physical system still requires domain knowledge in fluid dynamics, electromagnetism, or continuum mechanics.

Related Vector Calculus Concepts

Divergence and curl sit alongside other core operators in vector calculus:

• Gradient: Takes a scalar field (like temperature) and returns a vector field pointing in the direction of greatest increase.
• Line integrals: Integrate a vector field along a curve, useful for computing work done by a force.
• Flux integrals: Integrate the normal component of a vector field across a surface, closely related to divergence via the divergence theorem.

Using this divergence and curl calculator alongside gradient or flux tools can help build intuition for Maxwell’s equations, fluid flow problems, and many other topics in multivariable calculus.