Multivariable Calculus

Curl Calculator

Calculate the curl of a three-dimensional vector field symbolically. Find all six required partial derivatives, calculate each curl component, and optionally evaluate the curl at a specific point.

Calculate the Curl of a Vector Field

Enter the three components of a vector field \(\mathbf{F}(x,y,z)=\langle P,Q,R\rangle\).

\(P(x,y,z)\)
\(Q(x,y,z)\)
\(R(x,y,z)\)
Leave blank for a symbolic curl. Enter three coordinates to evaluate the curl at a point.
Supported notation: \(+\), \(-\), multiplication, division, powers, parentheses, implicit multiplication, and functions such as \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), \(\ln(x)\), \(e^x\), and \(\sqrt{x}\).

What is curl?

The curl of a vector field measures its local tendency to rotate around a point.

For a three-dimensional vector field

\[ \mathbf{F}(x,y,z) = \left\langle P,Q,R \right\rangle \]

its curl is

\[ \boxed{ \nabla\times\mathbf{F} = \left\langle R_y-Q_z,\; P_z-R_x,\; Q_x-P_y \right\rangle } \]

Physical meaning of curl

Curl is especially useful when a vector field represents fluid velocity.

  • The direction of curl gives the local axis of rotation.
  • The magnitude indicates the strength of the rotational tendency.
  • A zero curl indicates no local rotational component.

Curl is therefore closely related to rotational motion, circulation, and vorticity.

Curl Formula

The del operator is

\[ \nabla = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right\rangle. \]

For \(\mathbf F=\langle P,Q,R\rangle\), the curl is the cross product \(\nabla\times\mathbf F\):

\[ \boxed{ \nabla\times\mathbf F = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle } \]

The three components of the curl are often written as \(C_x,C_y,C_z\).

Example: Calculate the curl

Consider the vector field

\[ \mathbf F(x,y,z) = \left\langle -y,\; x,\; 0 \right\rangle. \]

Here

\[ P=-y,\qquad Q=x,\qquad R=0. \]

Calculate the required partial derivatives:

\[ R_y=0, \qquad Q_z=0 \] \[ P_z=0, \qquad R_x=0 \] \[ Q_x=1, \qquad P_y=-1. \]

Therefore,

\[ \nabla\times\mathbf F = \left\langle 0-0,\; 0-0,\; 1-(-1) \right\rangle \]
\[ \boxed{ \nabla\times\mathbf F = \left\langle 0,0,2 \right\rangle } \]

The positive \(z\)-direction indicates that the local rotational axis points along the positive \(z\)-axis.

What does zero curl mean?

If

\[ \nabla\times\mathbf F = \mathbf 0, \]

the vector field is called irrotational.

This means that the field has no local rotational tendency.

Curl and conservative fields

On an appropriate simply connected domain, a sufficiently smooth vector field with zero curl is associated with a conservative field.

In such situations there may exist a scalar potential \(f\) such that

\[ \mathbf F=\nabla f. \]

Applications of Curl

Fluid mechanics

When a vector field represents fluid velocity, curl is closely related to the local spinning motion of the fluid. The curl of velocity is commonly associated with vorticity.

Electromagnetism

Curl equations are fundamental to Maxwell's equations and describe relationships between electric and magnetic fields.

Vector calculus

Curl is one of the major differential operators together with gradient and divergence.

Circulation

Curl is connected to circulation through Stokes' theorem, which relates the circulation around a closed curve to the surface integral of curl.

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Frequently Asked Questions

The curl is a vector that measures the local rotational tendency of a three-dimensional vector field. For \(\mathbf F=\langle P,Q,R\rangle\), it is \[ \nabla\times\mathbf F = \langle R_y-Q_z,\, P_z-R_x,\, Q_x-P_y \rangle. \]

Calculate the six relevant partial derivatives: \(R_y,Q_z,P_z,R_x,Q_x,P_y\). Then combine them according to \[ \langle R_y-Q_z,\, P_z-R_x,\, Q_x-P_y \rangle. \]

Zero curl means the vector field has no local rotational component. Such a field is called irrotational.

Curl is a vector. Its direction describes the local rotational axis and its magnitude describes the strength of the rotational tendency.

Yes. Enter a point such as 1,2,3. The calculator first determines the symbolic curl and then evaluates its three components at the selected point.

Explore Vector Calculus

Curl is one of the fundamental differential operators in multivariable calculus. Continue exploring related Summe.org calculators for the gradient, directional derivative, divergence, Jacobian, Hessian, and critical points.

Gradient · Directional Derivative · Divergence · Jacobian · Hessian · Critical Points