Vector Fields

What we ultimately want to treat is the calculus of general functions \(\mathbb{R}^m\rightarrow\mathbb{R}^n\). We prepared for this first by raising the dimension of the codomain in §Curves and Vector-Valued Functions, and then the dimension of the domain after §Functions of Several Variables and Partial Derivatives. Now we combine both directions into one and begin the general case where both the domain and codomain are multidimensional. In particular, \(\mathbb{R}^n\rightarrow\mathbb{R}^n\), where the domain and codomain have the same dimension, is the most natural object; then this function takes an \(n\)-dimensional vector and outputs an \(n\)-dimensional vector.

Vector Fields

However, in practice, since the cross product, one of the powerful operations at our disposal, is defined only in \(3\) dimensions, we will conduct most of our discussion in \(3\) dimensions, and in its subspace, \(2\) dimensions. In any case, the following definitions make sense in general dimensions as well.

Definition 1 A function on a domain \(D \subseteq \mathbb{R}^n\) that assigns to each point \(\mathbf{x}\) a vector \(\mathbf{F}(\mathbf{x}) \in \mathbb{R}^n\), \(\mathbf{F}\colon D \rightarrow \mathbb{R}^n\), is called a vector field. In the plane we write \(\mathbf{F}(x,y) = (P(x,y), Q(x,y))\), and in space \(\mathbf{F}(x,y,z) = (P, Q, R)\); if each component \(P, Q, R\) is \(C^1\), we call \(\mathbf{F}\) a \(C^1\) vector field.

A vector field is most intuitively visualized as a picture of arrows attached to each point, originating from that point. For instance, in fluid flow, representing the velocity at each point forms a vector field. We already know one such object. (§Functions of Several Variables and Partial Derivatives, ⁋Definition 2)

Definition 2 For a \(C^1\) scalar field \(f\), the vector field given by the gradient \(\nabla f = (\partial f/\partial x_1, \ldots, \partial f/\partial x_n)\) is called the gradient field of \(f\). If, for some scalar field \(f\), a vector field can be written as \(\mathbf{F} = \nabla f\), the vector field \(\mathbf{F}\) is called a conservative field, and that \(f\) is called a potential of \(\mathbf{F}\).

Divergence and Curl

On the other hand, since not every vector field is a conservative field, determining whether a given \(\mathbf{F}\) is the gradient of some \(f\) becomes a central problem. The operations used for this determination are given in the following definition; in particular, because defining \(\curl\) requires the cross product, we inevitably have to come down to \(3\) dimensions.

Definition 3 The divergence of a \(C^1\) vector field \(\mathbf{F} = (P, Q, R)\) is the scalar field

\[\divergence \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]

and the curl is the vector field

\[\curl \mathbf{F} = \nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\]

A plane vector field \(\mathbf{F} = (P, Q)\) is viewed as independent of \(z\) in the form \((P, Q, 0)\), so \(\curl \mathbf{F} = (0, 0, \partial Q/\partial x - \partial P/\partial y)\), and this last component is called the curl of the plane vector field.

The two notations \(\nabla\cdot \mathbf{F}\) and \(\nabla \times \mathbf{F}\) above are formal, viewing \(\nabla = (\partial_x, \partial_y, \partial_z)\) as a vector and taking the dot and cross products.

Example 4 The radially extending \(\mathbf{F}(x,y,z) = (x, y, z)\) has \(\divergence \mathbf{F} = 1 + 1 + 1 = 3\), which is positive everywhere. Intuitively, this calculation tells us that this vector field points outward at every point. On the other hand, \(\curl \mathbf{F} = 0\). This can be understood as meaning that \(\mathbf{F}\) has no rotational component whatsoever and consists purely of an outward component.

As a contrasting example, \(\mathbf{G}(x,y,z) = (-y, x, 0)\) rotating around the \(z\)-axis has \(\divergence \mathbf{G} = 0\), but \(\curl \mathbf{G} = (0, 0, 2)\), showing that it is a vector field rotating about the \(z\)-axis.

Now let us see how these determine whether a given vector field is conservative. For this, we first need the following.

Proposition 5 For a \(C^2\) function \(f\) and a \(C^2\) vector field \(\mathbf{F}\),

\[\curl(\nabla f) = 0, \qquad \divergence(\curl \mathbf{F}) = 0\]

Proof

The first component of the curl of \(\nabla f = (f_x, f_y, f_z)\) is \(\partial_y f_z - \partial_z f_y = f_{zy} - f_{yz}\), which is \(0\) by §Functions of Several Variables and Partial Derivatives, ⁋Theorem 7 (Clairaut), and the remaining two components are \(0\) for the same reason. Also, the divergence of \(\curl \mathbf{F} = (R_y - Q_z,\ P_z - R_x,\ Q_x - P_y)\) groups as

\[\partial_x(R_y - Q_z) + \partial_y(P_z - R_x) + \partial_z(Q_x - P_y) = (R_{yx} - R_{xy}) + (P_{zy} - P_{yz}) + (Q_{xz} - Q_{zx})\]

and applying §Functions of Several Variables and Partial Derivatives, ⁋Theorem 7 (Clairaut) to each parenthesis again shows that it is \(0\).

The first identity gives a necessary condition for determining whether a vector field is conservative. This is because if \(\mathbf{F} = \nabla f\), then \(\curl \mathbf{F} = \curl(\nabla f) = 0\), so a vector field whose curl is not \(0\) can never be conservative. That is, the following holds.

Proposition 6 A conservative field is irrotational. That is, if a \(C^1\) vector field \(\mathbf{F}\) is conservative, then \(\curl \mathbf{F} = 0\). For a plane vector field \(\mathbf{F} = (P, Q)\), this is equivalent to \(\partial Q/\partial x = \partial P/\partial y\).

However, this condition is only a necessary condition, and the converse does not hold. The interesting point is that this converse depends on the shape of the domain: if the domain has no “holes,” then the converse also holds, but if there are holes, there exist vector fields that are irrotational and yet not conservative.

Example 7 Consider \(\mathbf{F} = (2xy,\ x^2 + z,\ y)\). Computing its curl,

\[\curl \mathbf{F} = (\partial_y y - \partial_z(x^2+z),\ \partial_z(2xy) - \partial_x y,\ \partial_x(x^2+z) - \partial_y(2xy)) = (1 - 1,\ 0,\ 2x - 2x) = 0\]

so it may be conservative. Therefore, let us see whether we can find an \(f\) satisfying \(\mathbf{F}=\nabla f\).

Such an \(f\) must first satisfy \(f_x = 2xy\) by the first component, so it must take the form \(f = x^2 y + g(y, z)\). Now differentiating this with respect to \(y\) gives \(f_y = x^2 + g_y\), and if \(\mathbf{F}\) were conservative, this value would have to match the second component \(x^2 + z\) of \(\mathbf{F}\), so \(g_y = z\), that is, \(g = yz + h(z)\). Now finally, substituting this back into \(f\), differentiating with respect to \(z\), and matching coefficients, we must have \(f_z = y + h'(z) = y\), so \(h' = 0\). Thus \(f = x^2 y + yz\) can serve as a potential, and indeed \(\mathbf{F} = \nabla f\) holds.

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References

[Ste] J. Stewart, Calculus, 8th ed., Cengage Learning, 2016.
[Kim] 김홍종, 미적분학 1·2, 제3개정판, 서울대학교출판문화원, 2020.