Curl (mathematics)

Representation in different coordinate systems

In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived.

In Cartesian coordinate systems

The notation ${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ${\displaystyle \mathrm{\nabla }}$ is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra.

Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ is, for ${\displaystyle \mathbf{F}}$ composed of ${\displaystyle [F_x,F_y,F_z]}$ (where the subscripts indicate the components of the vector, not partial derivatives): $${\displaystyle \mathrm{\nabla }\times \mathbf{F}=\left|\begin{array}{ccc}\hat{\mathit{ı}}& \hat{\mathit{ȷ}}& \hat{\mathit{k}}\\ {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}\\ F_x& F_y& F_z\end{array}\right|}$$ where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows: $${\displaystyle \mathrm{\nabla }\times \mathbf{F}=\left(\frac{\mathrm{\partial }{F}_z}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{F}_y}{\mathrm{\partial }z}\right)\hat{\mathit{ı}}+\left(\frac{\mathrm{\partial }{F}_x}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{F}_z}{\mathrm{\partial }x}\right)\hat{\mathit{ȷ}}+\left(\frac{\mathrm{\partial }{F}_y}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{F}_x}{\mathrm{\partial }y}\right)\hat{\mathit{k}}}$$

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

In orthogonal coordinate systems

The curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates can be calculated using the following expressions: $${\displaystyle \begin{array}{cc}& (\mathrm{curl}\mathbf{F}{)}_1=\frac{1}{h_2{h}_3}\left(\frac{\mathrm{\partial }(h_3{F}_3)}{\mathrm{\partial }{u}_2}-\frac{\mathrm{\partial }(h_2{F}_2)}{\mathrm{\partial }{u}_3}\right),\\ & (\mathrm{curl}\mathbf{F}{)}_2=\frac{1}{h_3{h}_1}\left(\frac{\mathrm{\partial }(h_1{F}_1)}{\mathrm{\partial }{u}_3}-\frac{\mathrm{\partial }(h_3{F}_3)}{\mathrm{\partial }{u}_1}\right),\\ & (\mathrm{curl}\mathbf{F}{)}_3=\frac{1}{h_1{h}_2}\left(\frac{\mathrm{\partial }(h_2{F}_2)}{\mathrm{\partial }{u}_1}-\frac{\mathrm{\partial }(h_1{F}_1)}{\mathrm{\partial }{u}_2}\right).\end{array}}$$ Equivalently, this can be written in determinant form: $${\displaystyle \begin{array}{c}\mathrm{\nabla }\times \mathbf{F}=\frac{1}{h_1{h}_2{h}_3}\left|\begin{array}{ccc}{h}_1{\hat{\mathbf{e}}}_1& h_2{\hat{\mathbf{e}}}_2& h_3{\hat{\mathbf{e}}}_3\\ {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{u}_1}}& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{u}_2}}& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{u}_3}}\\ h_1{\hat{F}}_1& h_2{\hat{F}}_2& h_3{\hat{F}}_3\end{array}\right|.\end{array}}$$

If (x₁, x₂, x₃) are the Cartesian coordinates and (u₁, u₂, u₃) are the orthogonal coordinates, then $${\displaystyle h_i=\sqrt{{\left(\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{u}_i}\right)}^2+{\left(\frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{u}_i}\right)}^2+{\left(\frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{u}_i}\right)}^2}}$$ is the length of the coordinate vector corresponding to u_i.

For convenience the h_i for commonly used orthogonal coordinates are tabulated below:

See orthogonal coordinates#Table of three-dimensional orthogonal coordinates for a more complete list.

In general (including non-orthogonal coordinate systems)

In a general coordinate system, the curl is given by $${\displaystyle (\mathrm{\nabla }\times \mathbf{F}{)}^k=\frac{1}{\sqrt{g}}{\epsilon }^{k\ell m}{\mathrm{\nabla }}_{\ell }{F}_m}$$ where ε denotes the Levi-Civita tensor, ∇ the covariant derivative, ${\displaystyle g}$ is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: $${\displaystyle (\mathrm{\nabla }\times \mathbf{F})=\frac{1}{\sqrt{g}}{\mathbf{R}}_k{\epsilon }^{k\ell m}{\mathrm{\partial }}_{\ell }{F}_m}$$ where R_k are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: $${\displaystyle \mathrm{\nabla }\times \mathbf{F}={\left(\star (\mathrm{d}({\mathbf{F}}^{\mathrm{♭}}))\right)}^{\mathrm{♯}}}$$

Here ♭ and ♯ are the musical isomorphisms, and ★ is the Hodge star operator. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.

Examples

Example 1

Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane (for z-axis component of the curl), zx-plane (for y-axis component of the curl) and yz-plane (for x-axis component of the curl vector). This can be seen in the examples below.

Example 2

The vector field $${\displaystyle \mathbf{F}(x,y,z)=y\hat{\mathit{ı}}-x\hat{\mathit{ȷ}}}$$ can be decomposed as $${\displaystyle F_x=y,F_y=-x,F_z=0.}$$

Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed.

Calculating the curl: $${\displaystyle \mathrm{\nabla }\times \mathbf{F}=0\hat{\mathit{ı}}+0\hat{\mathit{ȷ}}+\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x}(-x)-\frac{\mathrm{\partial }}{\mathrm{\partial }y}y\right)\hat{\mathit{k}}=-2\hat{\mathit{k}}}$$

The resulting vector field describing the curl would at all points be pointing in the negative z direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.

Example 3

For the vector field $${\displaystyle \mathbf{F}(x,y,z)=-x^2\hat{\mathit{ȷ}}}$$

the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line x = 3, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative z direction. Inversely, if placed on x = −3, the object would rotate counterclockwise and the right-hand rule would result in a positive z direction.

Calculating the curl: $${\displaystyle \mathrm{\nabla }\times \mathbf{F}=0\hat{\mathit{ı}}+0\hat{\mathit{ȷ}}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(-x^2\right)\hat{\mathit{k}}=-2x\hat{\mathit{k}}.}$$

The curl points in the negative z direction when x is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane x = 0.

Further examples

• In a vector field describing the linear velocities of each part of a rotating disk in uniform circular motion, the curl has the same value at all points, and this value turns out to be exactly two times the vectorial angular velocity of the disk (oriented as usual by the right-hand rule). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the vorticity of the flow at that point) equal to exactly two times the local vectorial angular velocity of the mass about the point.
• For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net torque on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the curl of the force field over the whole volume.
• Of the four Maxwell's equations, two—Faraday's law and Ampère's law—can be compactly expressed using curl. Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field.

Identities

In this section, all scalars and vector fields are assumed to be smooth, that is to be continuously differentiable of enough orders for the identity to make sense.

The curl of a cross product of vector fields v and F is $${\displaystyle \mathrm{\nabla }\times \left(\mathbf{v}\times \mathbf{F}\right)=(\left(\mathrm{\nabla }\cdot \mathbf{F}\right)+\mathbf{F}\cdot \mathrm{\nabla })\mathbf{v}-(\left(\mathrm{\nabla }\cdot \mathbf{v}\right)+\mathbf{v}\cdot \mathrm{\nabla })\mathbf{F}.}$$

Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: $${\displaystyle \mathbf{v}\times \left(\mathrm{\nabla }\times \mathbf{F}\right)={\mathrm{\nabla }}_{\mathbf{F}}\left(\mathbf{v}\cdot \mathbf{F}\right)-\left(\mathbf{v}\cdot \mathrm{\nabla }\right)\mathbf{F},}$$ where ∇_F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space).

Another example is the curl of a curl of a vector field. It can be shown that in general coordinates $${\displaystyle \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{F}\right)=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{F})-{\mathrm{\nabla }}^2\mathbf{F},}$$ and this identity defines the vector Laplacian of F, symbolized as ∇²F.

The curl of a gradient is always the zero vector field $${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\phi )=\mathbf{0}}$$ which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives.

The divergence of the curl of any vector field is equal to zero: $${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{F})=0.}$$

If φ is a scalar valued function and F is a vector field, then $${\displaystyle \mathrm{\nabla }\times (\phi \mathbf{F})=\mathrm{\nabla }\phi \times \mathbf{F}+\phi \mathrm{\nabla }\times \mathbf{F}}$$

Generalizations

The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$, these all being 3-dimensional spaces.

Differential forms

In 3 dimensions, a differential 0-form is a real-valued function ${\displaystyle f(x,y,z)}$; a differential 1-form is the following expression, where the coefficients are functions: $${\displaystyle a_{1}dx+a_{2}dy+a_{3}dz;}$$ a differential 2-form is the formal sum, again with function coefficients: $${\displaystyle a_{12}dx\wedge dy+a_{13}dx\wedge dz+a_{23}dy\wedge dz;}$$ and a differential 3-form is defined by a single term with one function as coefficient: $${\displaystyle a_{123}dx\wedge dy\wedge dz.}$$ (Here the a-coefficients are real functions of three variables; the wedge products, e.g. ${\displaystyle \text{d}x\wedge \text{d}y}$, can be interpreted as oriented plane segments, ${\displaystyle \text{d}x\wedge \text{d}y=-\text{d}y\wedge \text{d}x}$, etc.)

The exterior derivative of a k-form in R³ is defined as the (k + 1)-form from above—and in Rⁿ if, e.g., $${\displaystyle {\omega }^{(k)}=\sum _{1\le i_1<i_2<\cdots <i_k\le n}{a}_{i_1,\dots ,i_k}d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_k},}$$ then the exterior derivative d leads to $${\displaystyle d{\omega }^{(k)}=\sum _{\genfrac{}{}{0ex}{}{j=1}{i_1<\cdots <i_k}}^n\frac{\mathrm{\partial }{a}_{i_1,\dots ,i_k}}{\mathrm{\partial }{x}_j}d{x}_j\wedge d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_k}.}$$

The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, $${\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_j\mathrm{\partial }{x}_i},}$$ and antisymmetry, $${\displaystyle d{x}_i\wedge d{x}_j=-d{x}_j\wedge d{x}_i}$$

the twofold application of the exterior derivative yields ${\displaystyle 0}$ (the zero ${\displaystyle k+2}$-form).

Thus, denoting the space of k-forms by ${\displaystyle {\mathrm{\Omega }}^k({\mathbb{R}}^3)}$ and the exterior derivative by d one gets a sequence: $${\displaystyle 0\stackrel{d}{⟶}{\mathrm{\Omega }}^0\left({\mathbb{R}}^3\right)\stackrel{d}{⟶}{\mathrm{\Omega }}^1\left({\mathbb{R}}^3\right)\stackrel{d}{⟶}{\mathrm{\Omega }}^2\left({\mathbb{R}}^3\right)\stackrel{d}{⟶}{\mathrm{\Omega }}^3\left({\mathbb{R}}^3\right)\stackrel{d}{⟶}0.}$$

Here ${\displaystyle {\mathrm{\Omega }}^k({\mathbb{R}}^n)}$ is the space of sections of the exterior algebra ${\displaystyle {\mathrm{\Lambda }}^k({\mathbb{R}}^n)}$ vector bundle over Rⁿ, whose dimension is the binomial coefficient ${\displaystyle (\genfrac{}{}{0ex}{}{n}{k})}$; note that ${\displaystyle {\mathrm{\Omega }}^k({\mathbb{R}}^3)=0}$ for ${\displaystyle k>3}$ or ${\displaystyle k<0}$. Writing only dimensions, one obtains a row of Pascal's triangle:

$${\displaystyle 0\to 1\to 3\to 3\to 1\to 0;}$$

the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div.

Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-forms can be identified with k-vector fields (k-forms are k-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between k-vectors and (n − k)-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange k-forms, k-vector fields, (n − k)-forms, and (n − k)-vector fields; this is known as Hodge duality. Concretely, on R³ this is given by:

• 1-forms and 1-vector fields: the 1-form a_x dx + a_y dy + a_z dz corresponds to the vector field (a_x, a_y, a_z).
• 1-forms and 2-forms: one replaces dx by the dual quantity dy ∧ dz (i.e., omit dx), and likewise, taking care of orientation: dy corresponds to dz ∧ dx = −dx ∧ dz, and dz corresponds to dx ∧ dy. Thus the form a_x dx + a_y dy + a_z dz corresponds to the "dual form" a_z dx ∧ dy + a_y dz ∧ dx + a_x dy ∧ dz.

Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields:

• grad takes a scalar field (0-form) to a vector field (1-form);
• curl takes a vector field (1-form) to a pseudovector field (2-form);
• div takes a pseudovector field (2-form) to a pseudoscalar field (3-form)

On the other hand, the fact that d² = 0 corresponds to the identities $${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }f)=\mathbf{0}}$$ for any scalar field f, and $${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{v})=0}$$ for any vector field v.

Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and n-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and (n − 1)-forms are always fiberwise n-dimensional and can be identified with vector fields.

Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are

0 → 1 → 4 → 6 → 4 → 1 → 0;

so the curl of a 1-vector field (fiberwise 4-dimensional) is a 2-vector field, which at each point belongs to 6-dimensional vector space, and so one has $${\displaystyle {\omega }^{(2)}=\sum _{i<k=1,2,3,4}{a}_{i,k}d{x}_i\wedge d{x}_k,}$$ which yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero (d² = 0). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way.

However, one can define a curl of a vector field as a 2-vector field in general, as described below.

Curl geometrically

2-vectors correspond to the exterior power Λ²V; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{o}(V)}$ of infinitesimal rotations. This has (ⁿ
₂) = ⁠1/2⁠n(n − 1) dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have n = ⁠1/2⁠n(n − 1), which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a scalar field, as 2-dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{o}(4)}$.

The curl of a 3-dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page.

Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions.

Inverse

In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W) on a simply connected domain. This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential.

If W is a vector field with curl(W) = V, then adding any gradient vector field grad(f) to W will result in another vector field W + grad(f) such that curl(W + grad(f)) = V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart law.