What is the gradient of a scalar field?

The gradient of a scalar field f(x,y,z) is the vector of its partial derivatives: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). It points in the direction of steepest increase of f and its magnitude is the rate of maximum increase.

What does divergence measure physically?

The divergence of a vector field F measures the net outflow per unit volume at a point. Positive divergence means the point behaves like a source, negative divergence like a sink, and zero divergence indicates locally volume-preserving flow.

What does curl measure physically?

The curl of a vector field F measures the local rotation or swirling of the field. Its direction is the axis of rotation (given by the right-hand rule) and its magnitude is proportional to the angular speed of that rotation.

Can a vector field have zero divergence but nonzero curl?

Yes. An incompressible fluid flow can have zero divergence (no net sources or sinks) while still swirling, so its curl is nonzero. A classic example is a steady vortex flow around an axis.

Can a vector field have zero curl but nonzero divergence?

Yes. A purely radial field like F(x,y,z) = (x, y, z) has zero curl but positive divergence everywhere. It behaves like a uniform source with no local rotation.

Gradient, Divergence, and Curl Calculator with Step-by-Step Vector Calculus

Gradient, divergence, and curl: definitions

Gradient, divergence, and curl are the three core differential operators in vector calculus. They describe how scalar and vector fields change in space.

Gradient of a scalar field

For a scalar field \( f(x,y,z) \), the gradient is a vector field defined by

\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right). \]

\(\nabla f\) points in the direction of steepest increase of \(f\).
\(\|\nabla f\|\) is the maximum rate of change of \(f\) at that point.
Level surfaces \(f(x,y,z)=\text{const}\) are always orthogonal to \(\nabla f\).

Divergence of a vector field

For a vector field \( \mathbf{F}(x,y,z) = (F_1, F_2, F_3) \), the divergence is a scalar field

\[ \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}. \]

Physically, divergence measures net outflow per unit volume at a point.
\(\nabla \cdot \mathbf{F} > 0\): the point behaves like a source.
\(\nabla \cdot \mathbf{F} < 0\): the point behaves like a sink.
\(\nabla \cdot \mathbf{F} = 0\): the field is divergence-free (locally volume-preserving), e.g. incompressible fluid flow or magnetic fields.

Curl of a vector field

For a vector field \( \mathbf{F}(x,y,z) = (F_1, F_2, F_3) \), the curl is another vector field

\[ \nabla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right). \]

Curl measures the local rotation or “swirl” of the vector field.
The direction of \(\nabla \times \mathbf{F}\) is the axis of rotation (right-hand rule).
The magnitude \(\|\nabla \times \mathbf{F}\|\) is proportional to the angular speed of that rotation.
If \(\nabla \times \mathbf{F} = \mathbf{0}\), the field is called irrotational.

2D vs 3D formulas

In 2D, we usually work with variables \(x, y\) and treat the field as living in the plane.

2D gradient

\[ f(x,y) \quad\Rightarrow\quad \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} ). \]

2D divergence

\[ \mathbf{F}(x,y) = (P(x,y), Q(x,y)) \quad\Rightarrow\quad \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}. \]

2D curl (scalar “out of plane”)

In 2D, the curl of a planar vector field is a scalar representing the component of the 3D curl in the \(z\)-direction:

\[ \mathbf{F}(x,y) = (P(x,y), Q(x,y)) \quad\Rightarrow\quad \text{curl}_z(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}. \]

This scalar is often interpreted as the vorticity perpendicular to the plane.

Worked examples

Example 1: Gradient of a quadratic potential

Let \( f(x,y,z) = x^2 + y^2 + z^2 \). Then

\[ \frac{\partial f}{\partial x} = 2x,\quad \frac{\partial f}{\partial y} = 2y,\quad \frac{\partial f}{\partial z} = 2z, \] \[ \nabla f = (2x, 2y, 2z). \]

At the point \((1,2,3)\),

\[ \nabla f(1,2,3) = (2, 4, 6). \]

The gradient points radially outward and grows linearly with distance from the origin.

Example 2: Divergence and curl of a radial field

Consider the vector field \( \mathbf{F}(x,y,z) = (x, y, z) \).

Divergence:

\[ \nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3. \]

The divergence is constant and positive: the field behaves like a uniform source everywhere.

Curl:

\[ \nabla \times \mathbf{F} = \left( \frac{\partial z}{\partial y} - \frac{\partial y}{\partial z}, \frac{\partial x}{\partial z} - \frac{\partial z}{\partial x}, \frac{\partial y}{\partial x} - \frac{\partial x}{\partial y} \right) = (0 - 0,\; 0 - 0,\; 0 - 0) = (0,0,0). \]

The field has zero curl: it has no local rotation, only radial expansion.

Example 3: Pure rotation field

Take the 2D vector field \( \mathbf{F}(x,y) = (-y, x) \), which represents rotation around the origin.

Divergence:

\[ \nabla \cdot \mathbf{F} = \frac{\partial (-y)}{\partial x} + \frac{\partial x}{\partial y} = 0 + 0 = 0. \]

There are no sources or sinks: the flow is incompressible.

2D curl (out-of-plane component):

\[ \text{curl}_z(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \frac{\partial x}{\partial x} - \frac{\partial (-y)}{\partial y} = 1 - (-1) = 2. \]

The curl is a positive constant, indicating uniform counterclockwise rotation.

Key vector calculus identities

Some important identities involving gradient, divergence, and curl:

Gradient of a scalar multiple: \(\nabla (af) = a \nabla f\) for constant \(a\).
Divergence of a gradient (Laplacian): \(\nabla \cdot (\nabla f) = \nabla^2 f\).
Curl of a gradient: \(\nabla \times (\nabla f) = \mathbf{0}\) (gradients are irrotational).
Divergence of a curl: \(\nabla \cdot (\nabla \times \mathbf{F}) = 0\) (curls are divergence-free).

These identities underpin many results in physics, such as Maxwell’s equations in electromagnetism and the Navier–Stokes equations in fluid dynamics.

FAQ

How do I enter functions in this calculator?

Use standard programming-style syntax:

Powers: x^2, y^3, (x^2 + y^2)^(1/2)
Functions: sin(x), cos(y), tan(z), exp(x*y), ln(x), sqrt(x^2 + y^2)
Constants: pi, e

When should I use gradient vs divergence vs curl?

Use the gradient when you have a scalar field (e.g. temperature, potential) and want the direction and rate of steepest change.
Use divergence when you have a vector field (e.g. fluid velocity, electric field) and want to detect sources and sinks.
Use curl when you have a vector field and want to measure local rotation or vorticity.

What is the Laplacian and how is it related?

The Laplacian of a scalar field is defined as

\[ \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}. \]

It measures how \(f\) compares to its average value in a small neighborhood and appears in diffusion, heat conduction, and wave equations.

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