Question

If two functions of one variable, \(f\) and \(g\), have the property that \(f^{\prime}=g^{\prime},\) then \(f\) and \(g\) differ by a constant. Prove or disprove: If \(\mathbf{F}\) and \(\mathbf{G}\) are nonconstant vector fields in \(\mathbb{R}^{2}\) with curl \(\mathbf{F}=\operatorname{curl} \mathbf{G}\) and \(\operatorname{div} \mathbf{F}=\operatorname{div} \mathbf{G}\) at all points of \(\mathbb{R}^{2},\) then \(\mathbf{F}\) and \(\mathbf{G}\) differ by a constant vector.

Short answer

Short Answer: The statement "if the curl and divergence of two vector fields, \(\mathbf{F}\) and \(\mathbf{G}\), are equal, then they must differ only by a constant vector" is incorrect. In general, if \(\mathbf{F}\) and \(\mathbf{G}\) are nonconstant vector fields in \(\mathbb{R}^{2}\) with curl \(\mathbf{F}=\operatorname{curl} \mathbf{G}\) and with \(\operatorname{div} \mathbf{F}=\operatorname{div} \mathbf{G}\) at all points of \(\mathbb{R}^{2},\) then \(\mathbf{F}\) and \(\mathbf{G}\) can differ by a position-dependent vector field and not necessarily a constant vector.

Step-by-step solution

Write down the curl and divergence of \(\mathbf{F}\) and \(\mathbf{G}\)

First, let's write down the curl and divergence of the two vector fields \(\mathbf{F}\) and \(\mathbf{G}\). In \(\mathbb{R}^{2}\), these are given by, for \(\mathbf{F} = F_1 \mathbf{i} + F_2 \mathbf{j}\) and \(\mathbf{G} = G_1 \mathbf{i} + G_2 \mathbf{j}\), as: $$ \operatorname{curl} \mathbf{F} = \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} $$ $$ \operatorname{curl} \mathbf{G} = \frac{\partial G_2}{\partial x} - \frac{\partial G_1}{\partial y} $$ $$ \operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} $$ $$ \operatorname{div} \mathbf{G} = \frac{\partial G_1}{\partial x} + \frac{\partial G_2}{\partial y} $$

Equate the curl and divergence of \(\mathbf{F}\) and \(\mathbf{G}\)

Since the problem states that \(\mathbf{F}\) and \(\mathbf{G}\) have the same curl and divergence, we can equate the above expressions to get two equations: $$ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} = \frac{\partial G_2}{\partial x} - \frac{\partial G_1}{\partial y} $$ $$ \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} = \frac{\partial G_1}{\partial x} + \frac{\partial G_2}{\partial y} $$

Analyze the equations

Now, let's see if we can find a relation between \(F_1\), \(F_2\), \(G_1\), and \(G_2\). Adding the two equations above, we get: $$ 2 \frac{\partial F_2}{\partial x} = 2 \frac{\partial G_2}{\partial x} $$ which simplifies to: $$ \frac{\partial F_2}{\partial x} = \frac{\partial G_2}{\partial x} $$ This implies: $$ F_2(x,y) - G_2(x,y) = C_1(y) $$ for some function \(C_1(y)\) of only the variable \(y\). Similarly, subtracting the two equations, we get: $$ 2 \frac{\partial F_1}{\partial y} = 2 \frac{\partial G_1}{\partial y} $$ which simplifies to: $$ \frac{\partial F_1}{\partial y} = \frac{\partial G_1}{\partial y} $$ This implies: $$ F_1(x,y) - G_1(x,y) = C_2(x) $$ for some function \(C_2(x)\) of only the variable \(x\).

Conclusion

From the above analysis, we can see that the difference between \(\mathbf{F}\) and \(\mathbf{G}\) can generally be represented as: $$ \mathbf{F} - \mathbf{G} = C_2(x) \mathbf{i} + C_1(y) \mathbf{j} $$ This shows that \(\mathbf{F}\) and \(\mathbf{G}\) do not necessarily differ by a constant vector, but by a general position-dependent vector field. Thus, we can disprove the statement given in the exercise. If \(\mathbf{F}\) and \(\mathbf{G}\) are nonconstant vector fields in \(\mathbb{R}^{2}\) with curl \(\mathbf{F}=\operatorname{curl} \mathbf{G}\) and \(\operatorname{div} \mathbf{F}=\operatorname{div} \mathbf{G}\) at all points of \(\mathbb{R}^{2},\) then \(\mathbf{F}\) and \(\mathbf{G}\) do not necessarily differ by a constant vector.

Key concepts

Curl of a Vector Field

Imagine the wind as it whirls around in a storm; Mathematicians capture this rotation using a concept called the curl of a vector field. The curl measures the tendency of the vector field to rotate around a point. In mathematical terms, the curl of a vector field \textbf{F} represented as \textbf{F} = F1 \textbf{i} + F2 \textbf{j} + F3 \textbf{k} in three dimensions, is given by the cross product of the del operator \textbf{\(abla\)} with the vector field \textbf{F}, that is:
\[\text{curl } \mathbf{F} = abla \times \mathbf{F}\].
In two dimensions, it simplifies to \[\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\].
This also implies that in a two-dimensional vector field, the curl will always point in the direction of the k vector (out of the x-y plane), effectively measuring the 'twist' of the field in the plane.

• Positive curl means the field tends to rotate counterclockwise.
• Negative curl indicates a clockwise rotation.
• A curl of zero suggests no rotation at all; the field is said to be irrotational.

Thus, if two vector fields in \textbf{\(\mathbb{R}^2\)} have the same curl at all points, they exhibit the same local rotational behavior. However, as seen in the example, equal curls do not necessarily mean the fields are identical.

Divergence of a Vector Field

The divergence of a vector field relates to how much the field 'diverges' from a given point, akin to how an inflating balloon causes air to spread out from its center. More formally, for a vector field \textbf{F} = F1 \textbf{i} + F2 \textbf{j} + F3 \textbf{k}, the divergence is the dot product of the del operator with the vector field, defined as:
\[\text{div } \mathbf{F} = abla \cdot \mathbf{F}\].
In two dimensions, this becomes \[\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}\].
The divergence gives us a measure of how much a vector field behaves like a source or a sink at a given point. Essentially,

• A positive divergence indicates a 'source', with vectors moving away from the point.
• A negative divergence suggests a 'sink', with vectors moving towards the point.
• Zero divergence means the vector field is 'sourceless' and 'sinkless', or incompressible.

When two vector fields have the same divergence at all points, the flow of the field is similar in the sense of spreading out or converging. However, as the initial problem demonstrates, identical divergence does not guarantee that the two fields are the same. It informs us about their local expansion or compression but not their exact structure or behavior.

Partial Derivatives in Vector Functions

Understanding the changes in a multi-variable function is key to unraveling complex dynamical systems, like weather patterns or fluid flow. Enter partial derivatives: they tell us how a function changes as one particular variable shifts, while others remain constant.
In the context of vector fields, which are functions that assign vectors to points in space, partial derivatives help us analyze these fields component by component. For a vector field \textbf{F} = F1 \textbf{i} + F2 \textbf{j} + F3 \textbf{k}, the partial derivative of F1 with respect to x, denoted \(\frac{\partial F_1}{\partial x}\), tells us how the x-component of \textbf{F} changes as we move along the x-axis.
This concept is instrumental in defining both the curl and divergence of vector fields, as we have seen:

• The curl involves the partial derivatives of the components of the field to capture rotational effects.
• The divergence uses partial derivatives to gauge how the field behaves like a source or a sink.

Vector fields can be tough to visualize, so considering how each component changes in space aids in constructing a complete picture of the field's behavior. It's important to note that while partial derivatives give insights into local variations of a vector field, they do not necessarily convey information about the field's global structure, as highlighted by the exercise problem. As we unravel each component's behavior, we gain the ability to understand and predict the overall behavior of the field.