Vector Fields

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

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

Vector Fields

However, since the cross product (one of the most powerful tools at our disposal) is defined only in three dimensions, we will for the most part work in three dimensions and its subspace, the plane. Nevertheless, the following definitions make sense in arbitrary dimension.

Definition 1 A function \(\mathbf{F}\colon D \rightarrow \mathbb{R}^n\) that assigns to each point \(\mathbf{x}\) in a domain \(D \subseteq \mathbb{R}^n\) a vector \(\mathbf{F}(\mathbf{x}) \in \mathbb{R}^n\) 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 emanating from each point. For instance, the flow velocity of a fluid at each point constitutes a vector field. We have already encountered one such object. (§Functions of Several Variables and Partial Derivatives, ⁋Definition 2)

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

Divergence and Curl

On the other hand, not every vector field is conservative, so the central problem becomes determining whether a given \(\mathbf{F}\) is the gradient of some \(f\). The operations used for this test are given in the following definition; in particular, because defining \(\curl\) requires the cross product, we are inevitably forced down to three 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 regarded as \((P, Q, 0)\) independent of \(z\), 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 with it.

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

In contrast, the vector field \(\mathbf{G}(x,y,z) = (-y, x, 0)\) 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 operations allow us to test 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 vanishes by §Functions of Several Variables and Partial Derivatives, ⁋Theorem 7 (Clairaut), and the remaining two components are zero for the same reason. Also, the divergence of \(\curl \mathbf{F} = (R_y - Q_z,\ P_z - R_x,\ Q_x - P_y)\) is

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

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

The first identity yields a necessary condition for conservativity. If \(\mathbf{F} = \nabla f\), then \(\curl \mathbf{F} = \curl(\nabla f) = 0\), so a vector field with nonzero curl 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 necessary, and the converse does not hold in general. The interesting point is that the 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 irrotational vector fields that are 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. 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 \(f = x^2 y + g(y, z)\). Differentiating this with respect to \(y\) gives \(f_y = x^2 + g_y\), and if \(\mathbf{F}\) were conservative this must match the second component \(x^2 + z\) of \(\mathbf{F}\), so \(g_y = z\), i.e., \(g = yz + h(z)\). Finally, substituting this back into \(f\) and differentiating with respect to \(z\) and matching coefficients, we need \(f_z = y + h'(z) = y\), 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.