What is curl in electromagnetics?

Thus, we are motivated to formally define circulation and then to figure out how to most conveniently apply the concept in mathematical analysis. The result is the curl operator. So, we begin with the concept of circulation:

Specifically, the circulation of the vector field \({\bf A}({\bf r})\) over the closed path \({\mathcal C}\) is

\[\oint_{\mathcal C} {\bf A}\cdot d{\bf l} \nonumber \]

The circulation of a uniform vector field is zero for any valid path. For example, the circulation of \({\bf A}=\hat{\bf x}A_0\) is zero because non-zero contributions at each point on \({\mathcal C}\) cancel out when summed over the closed path. On the other hand, the circulation of \({\bf A}=\hat{\bf \phi}A_0\) over a circular path of constant \(\rho\) and \(z\) is a non-zero constant, since the non-zero contributions to the integral at each point on the curve are equal and accumulate when summed over the path.

Curl is, in part, an answer to the question of what the circulation at a point in space is. In other words, what is the circulation as \({\mathcal C}\) shrinks to it’s smallest possible size. The answer in one sense is zero, since the arclength of \({\mathcal C}\) is zero in this limit – there is nothing to integrate over. However, if we ask instead what is the circulation per unit area in the limit, then the result should be the non-trivial value of interest. To express this mathematically, we constrain \({\mathcal C}\) to lie in a plane, and define \({\mathcal S}\) to be the open surface bounded by \({\mathcal C}\) in this case. Then, we define the scalar part of the curl of \({\bf A}\) to be:

\[\lim_{\Delta s \to 0} \frac{\oint_{\mathcal C} {\bf A}\cdot d{\bf l}}{\Delta s} \nonumber \]

where \(\Delta s\) is the area of \({\mathcal S}\), and (important!) we require \({\mathcal C}\) and \({\mathcal S}\) to lie in the plane that maximizes the above result.

Because \({\mathcal S}\) and it’s boundary \({\mathcal C}\) lie in a plane, it is possible to assign a direction to the result. The chosen direction is the normal \(\hat{\bf n}\) to the plane in which \({\mathcal C}\) and \({\mathcal S}\) lie. Because there are two normals at each point on a plane, we specify the one that satisfies the right hand rule. This rule, applied to the curl, states that the correct normal is the one which points through the plane in the same direction as the fingers of the right hand when the thumb of your right hand is aligned along \({\mathcal C}\) in the direction of integration. Why is this the correct orientation of \(\hat{\bf n}\)? See Section 4.9 for the answer to that question. For the purposes of this section, it suffices to consider this to be simply an arbitrary sign convention.

Now with the normal vector \(\hat{\bf n}\) unambiguously defined, we can now formally define the curl operation as follows:

\[\mbox{curl}~{\bf A} \triangleq \lim_{\Delta s \to 0} \frac{\hat{\bf n}\oint_{\mathcal C} {\bf A}\cdot d{\bf l}}{\Delta s} \label{m0048_eCurlDef} \]

where, once again, \(\Delta s\) is the area of \({\mathcal S}\), and we select \({\mathcal S}\) to lie in the plane that maximizes the magnitude of the above result. Summarizing:

Curl is a very important operator in electromagnetic analysis. However, the definition (Equation \ref{m0048_eCurlDef}) is usually quite difficult to apply. Remarkably, however, it turns out that the curl operation can be defined in terms of the \(\nabla\) operator; that is, the same \(\nabla\) operator associated with the gradient, divergence, and Laplacian operators. Here is that definition: \[\mbox{curl}~{\bf A} \triangleq \nabla \times {\bf A} \nonumber \] For example: In Cartesian coordinates,

\[\nabla \triangleq \hat{\bf x}\frac{\partial}{\partial x} + \hat{\bf y}\frac{\partial}{\partial y} + \hat{\bf z}\frac{\partial}{\partial z} \nonumber \]

and

\[{\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z \nonumber \]

so curl can be calculated as follows:

\[\nabla\times{\bf A} = \begin{vmatrix} \hat{\bf x} & \hat{\bf y} & \hat{\bf z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix} \nonumber \]

or, evaluating the determinant:

\[\begin{split} \nabla \times {\bf A} &= \hat{\bf x}\left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) \\ &~~ +\hat{\bf y}\left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) \\ &~~ +\hat{\bf z}\left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) \end{split} \nonumber \]

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.[1] The curl of a field is formally defined as the circulation density at each point of the field.

A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

Curl F is a notation common today to the United States and Americas. In many European countries, particularly in classic scientific literature, the alternative notation rot F is traditionally used, which is spelled as "rotor", and comes from the "rate of rotation", which it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator ∇ × F {\displaystyle \nabla \times \mathbf {F} } [2] which also reveals the relation between curl (rotor), divergence, and gradient operators.

Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation ∇ × {\displaystyle \nabla \times } for the curl.

The name "curl" was first suggested by James Clerk Maxwell in 1871[3] but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.[4][5]

The curl of a vector field F, denoted by curl F, or ∇ × F {\displaystyle \nabla \times \mathbf {F} } , or rot F, is an operator that maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3. It can be defined in several ways, to be mentioned below:

One way to define the curl of a vector field at a point is implicitly through its projections onto various axes passing through the point: if u ^ {\displaystyle \mathbf {\hat {u}} } is any unit vector, the projection of the curl of F onto u ^ {\displaystyle \mathbf {\hat {u}} } may be defined to be the limiting value of a closed line integral in a plane orthogonal to u ^ {\displaystyle \mathbf {\hat {u}} } divided by the area enclosed, as the path of integration is contracted indefinitely around the point.

More specifically, the curl is defined at a point p as[6][7]

where the line integral is calculated along the boundary C of the area A in question, |A| being the magnitude of the area. This equation defines the projection of the curl of F onto u ^ {\displaystyle \mathbf {\hat {u}} } . The infinitesimal surfaces bounded by C have u ^ {\displaystyle \mathbf {\hat {u}} } as their normal. C is oriented via the right-hand rule.

The above formula means that the projection of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field projected onto a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in the same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately.

To this definition fits naturally the Kelvin–Stokes theorem, as a global formula corresponding to the definition. It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface.

Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell is contracted indefinitely around p.

More specifically, the curl may be defined by the vector formula

where the surface integral is calculated along the boundary S of the volume V, |V| being the magnitude of the volume, and n ^ {\displaystyle \mathbf {\hat {n}} } pointing outward from the surface S perpendicularly at every point in S.

In this formula, the cross product in the integrand measures the tangential component of F at each point on the surface S, together with the orientation of these tangential components with respect to the surface S. Thus, the surface integral measures the overall extent to which F circulates around S, together with the net orientation of this circulation in space. The curl of a vector field at a point is then the infinitesimal volume density of the net vector circulation (i.e., both magnitude and spatial orientation) of the field around the point.

To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume.

Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates:

The equation for each component (curl F)k can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices).

If (x1, x2, x3) are the Cartesian coordinates and (u1, u2, u3) are the orthogonal coordinates, then

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 centre 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 centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.[8]

The curl of the vector 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 clearly seen in the examples below.

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.

The notation ∇ × F has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ 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),∇ × F is, for F composed of [Fx, Fy, Fz] (where the subscripts indicate the components of the vector, not partial derivatives):

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:[9]: 43

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

In a general coordinate system, the curl is given by[1]

where ε denotes the Levi-Civita tensor, ∇ the covariant derivative, g {\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:

where Rk are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as:

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.

The vector field

can be decomposed as

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:

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.

For the vector field

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:

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.

In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be

Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field:

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

and this identity defines the vector Laplacian of F, symbolized as ∇2F.

The curl of the gradient of any scalar field φ is always the zero vector field

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:

If φ is a scalar valued function and F is a vector field, then

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 s o {\displaystyle {\mathfrak {so}}} (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 s o {\displaystyle {\mathfrak {so}}} (3), these all being 3-dimensional spaces.

In 3 dimensions, a differential 0-form is simply a function f(x, y, z); a differential 1-form is the following expression, where the coefficients are functions:

a differential 2-form is the formal sum, again with function coefficients:

and a differential 3-form is defined by a single term with one function as coefficient:

(Here the a-coefficients are real functions of three variables; the "wedge products", e.g. dx ∧ dy, can be interpreted as some kind of oriented area elements, dx ∧ dy = −dy ∧ dx, 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.,

then the exterior derivative d leads to

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, e.g. because of

the twofold application of the exterior derivative leads to 0.

Thus, denoting the space of k-forms by Ωk(R3) and the exterior derivative by d one gets a sequence:

Here Ωk(Rn) is the space of sections of the exterior algebra Λk(Rn) vector bundle over Rn, whose dimension is the binomial coefficient (nk); note that Ωk(R3) = 0 for k > 3 or k < 0. Writing only dimensions, one obtains a row of Pascal's triangle:

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:

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

On the other hand, the fact that d2 = 0 corresponds to the identities

for any scalar field f, and

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

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

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.

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 s o {\displaystyle {\mathfrak {so}}} (V) of infinitesimal rotations. This has (n2) = 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 function, 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 s o ( 4 ) {\displaystyle {\mathfrak {so}}(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.[10]

In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W).[citation needed] 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.

Curl is an operation, which when applied to a vector field, quantifies the circulation of that field. The concept of circulation has several applications in electromagnetics.

where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via a limiting process and is the unit normal vector to this region. If , then the field is said to be an irrotational field. The symbol is variously known as "nabla" or "del."

The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in two of the four Maxwell equations,

where MKS units have been used here, denotes the electric field, is the magnetic field, is a constant of proportionality known as the permeability of free space, is the current density, and is another constant of proportionality known as the permittivity of free space. Together with the two other of the Maxwell equations, these formulas describe virtually all classical and relativistic properties of electromagnetism.

In Cartesian coordinates, the curl is defined by

This provides the motivation behind the adoption of the symbol for the curl, since interpreting as the gradient operator , the "cross product" of the gradient operator with is given by

which is precisely equation (4). A somewhat more elegant formulation of the curl is given by the matrix operator equation

(Abbott 2002).

The curl can be similarly defined in arbitrary orthogonal curvilinear coordinates using

and defining

as

The curl can be generalized from a vector field to a tensor field as

It is a vector whose magnitude is the maximum circulation of the given field per unit area (tending to zero) and whose direction is normal to the area when it is oriented for maximum circulation.

In simple words, the curl can be considered analogues to the circulation or whirling of the given vector field around the unit area. More are the field lines circulating along the unit area around the point, more will be the magnitude of the curl.

The direction of the curl vector gives us an idea of the nature of rotation. It always follows the right-hand thumb rule where the thumb denotes the direction of curl vector and finger denotes the way of maximum circulation of the unit area.

The curl of a vector field is the mathematical operation whose answer gives us an idea about the circulation of that field at a given point. In other words, it indicates the rotational ability of the vector field at that particular point.

Technically, it is a vector whose magnitude is the maximum circulation of the given field per unit area (tending to zero) and whose direction is normal to the area when it is oriented for maximum circulation.

Now if we want to find the product of the component of the field along the line at every point and length of the line then we take line integral i.e.

In simple words, the line integration would give us the effect of the vector field along the given line.

Now if I calculate the line integration of the given field along the path L, then in simple words, I would get the effect of the vector field along the L or boundary of the surface ‘ds‘. Now, what does this indicate? It is indicating how much the field is circulating the given area ‘ds‘. Isn’t it? It is represented as follows-

So this close line integration of the field around the boundary of the surface ‘ds’ is called as the circulation of the vector field. In layman’s words, it indicates the rotating or whirling capacity of the field if the surface is allowed to rotate. More is the circulation, more would be the answer of this integration.

Now again jump to the definition of the curl. It requires maximum circulation per unit area i.e. area should approach to zero. So, mathematically it can be written as follows

Now the final part of the definition that is the direction of the curl vector. According to the definition, it is normal to the area/surface such that the surface is aligned for the maximum possible circulation. Now, this can be easily determined using the right-hand thumb rule where thumb denoting the axis of the rotation if the surface is allowed to rotate according to the circulation of the field. And this will be the direction of the curl vector.

So mathematically, the definition would be as shown in the following figure where the bracketed term is the maximum circulation as discussed above and the unit vector according to the right-hand rule.

The curl, in simple words, is the rotating or whirling nature of the vector field at a given point. Again in simpler words, it can be explained as – Assume that I have put a small surface at that point in the hypothetical vector field similar to force. Let us also assume that the surface has a fixed center but flexible axis. Then, this surface can rotate about the particular axis. The position of the axis and magnitude at that point for the maximum rotation can be considered as the Curl of this vector field.

As I have assumed the vector field is similar to force, it would be able to rotate that imaginary surface. But actually, for any other vector field, we would say the circulation of the field lines around that surface. Isn’t it? More circulation more curl. And as I explained in the last article, the numerator of the above formula denotes the circulation.

Now according to the definition, the field lines of E will be circulating along the differential surface present at point P. And the axis of the curl will be decided using right-hand rule so that maximum circulation of the field is possible at that position. This position or orientation of the axis can be anywhere in the space. But for simplicity consider the axis along positive X-axis. Actually, the complete curl effect will be the combined effect (vector addition) of all three axes together i.e. along X, along Y and along Z.

I am assuming the differential surface dS i.e. infinitesimally small surface bounded by differential lengths. Hence the line integration at the given point can be thought as the multiplication of the field value i.e. E and the differential length at that point.

The length AB is the infinitesimally small length along Y-axis hence it can be considered as ‘dy’. The value of the function E at the location AB can be approximated using Taylor’s series.

I have explained in previous article, how any function can be expanded around the given point in terms of spatial derivatives. In this case, the y-component of the field E can be expressed with Taylor’s series expansion around P(x0, y0, z0) as shown below. I am considering only y component of the vector field E i.e. Ey for a reason. Can you guess why?

I have considered only Ey because I am considering the integration from A to B. So for this integration, I need the component of the field E along AB i.e. along Y-axis, hence only Ey . Now the given line AB is at distance (dz/2) below from the point P. Hence from the above formula, we can consider only the term consisting derivative w.r.t. z. Also, z is taken equal to [z0 – (dz/2)]. So the value of Ey at the location of the line AB is as follows –

So the integration of E along AB can be written for the differential case as follows. The Ey is given by the above equation and the dl is dy as stated initially.

You can easily work out in a similar manner to get the remaining integrals for BC, CD and DA. In each case, the value of the field E and dl are modified accordingly.

So for the complete close integration along ABCD i.e. circulation of the field, we need to add all these four terms together and we will get the following.

But, wait. Can you observe the term dydz. What does this represent? It is the area of the differential surface dS that we have assumed at the starting of our discussion. Isn’t it? Also as we have assumed this area to be very small, differentially small we can say it is approaching zero. So above relation can be written as follows

Bingo! Check out the left-hand side of the above equation. It is exactly the same as the definition of the Curl. Am I right? So this is the expression of the curl of our assumed vector field E having axis along X-axis (as we have assumed so initially).

On similar lines, we can proceed step by step as we did here and find the Y and Z components of the Curl.

Rather than these three formulas for different components, the complete Curl formula in matrix form is represented as follows. This matrix form formula after simplification would also lead to the above three formulas.

Let us consider the surface of the imaginary water current. It is the best example of the vector field.

After all, a vector field can be considered as the set of vectors present in the space defined by a certain function. In other words, each point in the field bears a vector whose magnitude and direction is decided by the given function. And the surface of the water current can be easily simulated as a vector field.

The velocities of the water current at different points of the surface form the set of vectors i.e. field as shown in the following figure. The magnitude and the directions of these individual vectors are governed by a certain function.

Again for simplicity, we can assume that the velocity vector is nothing but the force that is experienced by any point object placed at that point. So the above diagram of the velocity vector field becomes the force vector field. This way we can say that these force vectors are responsible for any movement of the object that is placed on the surface.

Now let us assume that a small piece of paper or lamina of area ‘ds’ is put on this water surface. For simplicity let us consider the ‘ds’ to be a square. Also, consider that this ‘ds’ is a hypothetical area which doesn’t flow along with the stream of water but remains stationary at the given point. Yes, but assume that it can rotate about its axis.

Now let me explain this by another way i.e. using the concepts of line integration. We know that the line integration of the vector field along any path is the sum of all the infinitesimal products consisting of the field value (parallel to the path) and the infinitesimal length.

Note the TWO points in above integration. First, the line integrations along BC and DA are both zero. The reason is that the angle between F and dl are 900 and 2700 respectively. And for the line integration, we want the component of the field along (parallel to) the path i.e. Fcosθ. So they both are zero.

Second, the integration for CD is negative. The reason for this is also the same, the angle θ. Here it is 1800. Both F and dl are going anti-parallel.

Now the line integration along the ABCDA can be called as the circulation of the field lines for the surface ‘ds’. In other words, this circulation i.e. the answer of this integration is responsible for the net circulation of the area ‘ds’.

In our case, this is a zero. So the rotational nature is zero i.e. the curl of the field is zero.

This would be similar to Case 2. In this case, also, the net circulation of the field is not zero and responsible for the circulation of the ds.

But one very important thing we can notice here that the nature of the rotation. In case 2 the nature of rotation was anticlockwise as F1 is greater than F2. But in this case, it will be clockwise. If we consider the axis of rotation according to the right-hand rule then the thumb direction is opposite in both of the cases.

We have modelled a vector field as a surface of the water current. This may not be 100% true as the actual field may occupy the 3D space.

So in case of the actual vector field, this Curl example should be stretched further to get the axis of the rotation of ‘ds’ such that maximum circulation is possible.