Question

Assume the vector field \(\mathbf{F}=\langle f, g\rangle\) is source free (zero divergence) with stream function \(\psi\). Let \(C\) be any smooth simple curve from \(A\) to the distinct point \(B\). Show that the flux integral \(\int_{C} \mathbf{F} \cdot \mathbf{n} d s\) is independent of path; that is, \(\int_{C} \mathbf{F} \cdot \mathbf{n} d s=\psi(B)-\psi(A)\).

Short answer

Answer: The flux integral of a source-free vector field along a curve is independent of the path and equals the difference in the stream function values at the end points, i.e., $\psi(B) - \psi(A)$.

Step-by-step solution

Find the relationship between the vector field components and the stream function

Since the vector field \(\mathbf{F}\) has a stream function \(\psi\), we can write \(\mathbf{F} = \langle f, g \rangle\) in terms of the gradient of the stream function as follows: \begin{align*} f &= -\frac{\partial \psi}{\partial y},\\ g &= \frac{\partial \psi}{\partial x}. \end{align*} This is because the streamlines of \(\mathbf{F}\) are the contours of the stream function \(\psi\), which implies that \(\mathbf{F}\) is orthogonal to the gradient of \(\psi\).

Compute the flux integral of \(\mathbf{F}\) along a curve

The flux integral of the vector field \(\mathbf{F}\) along the curve \(C\) can be written as \[\int_C \mathbf{F} \cdot \mathbf{n} ds,\] where \(\mathbf{n}\) is the outward-pointing unit normal vector to the curve. We have \[\mathbf{n} = \frac{\mathbf{r'}\times \mathbf{k}}{\left\Vert \mathbf{r'}\times \mathbf{k} \right\Vert},\] where \(\mathbf{k}\) is the unit vector perpendicular to the plane, and \(\mathbf{r'}\) is the tangent vector given by \[\mathbf{r'} = \frac{d\mathbf{r}}{ds} = \langle \frac{dx}{ds}, \frac{dy}{ds} \rangle.\] Now let's compute \(\mathbf{F} \cdot \mathbf{n}\) and express it in terms of \(\psi\). Since \(\mathbf{F} = \langle f, g \rangle = \left\langle -\frac{\partial\psi}{\partial y}, \frac{\partial\psi}{\partial x} \right\rangle\), we have \begin{align*} \mathbf{F}\cdot\mathbf{n} &= \left\langle -\frac{\partial\psi}{\partial y}, \frac{\partial\psi}{\partial x}\right\rangle \cdot \frac{\mathbf{r'}\times \mathbf{k}}{\left\Vert \mathbf{r'}\times \mathbf{k} \right\Vert}\\ &= \frac{\left\langle -\frac{\partial\psi}{\partial y}, \frac{\partial\psi}{\partial x}\right\rangle \cdot \left\langle \frac{dy}{ds}, -\frac{dx}{ds}\right\rangle}{\left\Vert \mathbf{r'}\times \mathbf{k} \right\Vert}\\ &= \frac{(-\frac{\partial\psi}{\partial y})(\frac{dy}{ds})+(\frac{\partial\psi}{\partial x})(-\frac{dx}{ds})}{\left(\frac{dx^2}{ds^2}+\frac{dy^2}{ds^2}\right)^{1/2}}\\ &= -\frac{d\psi}{ds}. \end{align*} So the flux integral can be written as \[-\int_C \frac{d\psi}{ds} ds.\]

Show that the flux integral is independent of the path

As we have shown above, the flux integral is expressed as \[-\int_C \frac{d\psi}{ds} ds.\] If we integrate this expression, we get \[-\int_C \frac{d\psi}{ds} ds = -\left[\psi(B) - \psi(A)\right],\] which is independent of the path taken from point \(A\) to point \(B\). Therefore, the flux integral is independent of path, and equals \(\psi(B) - \psi(A)\).

Key concepts

Stream Function

A stream function, often denoted as \( \psi \), is a powerful concept in fluid dynamics and vector calculus. It is used to describe the flow of a fluid, where the vector field \( \mathbf{F} = \langle f, g \rangle \) can be represented in terms of the stream function \( \psi \). The components of the vector field are defined as the gradients of \( \psi \). Specifically, the function \( \psi \) must satisfy the relationships:

• \( f = -\frac{\partial \psi}{\partial y} \)
• \( g = \frac{\partial \psi}{\partial x} \)

These equations ensure that \( \mathbf{F} \) is always orthogonal to the gradient of the stream function, aligning with the flow of the fluid. These orthogonal relationships mean that the streamlines in a flow can be interpreted directly from the contours of \( \psi \). Intuitively, this stream function helps visualize the flow pattern of a vector field by allowing us to understand how the field behaves at different points.

Flux Integral

Flux integrals are used to measure the quantity of a vector field passing through a given curve or surface. In our case, we are looking at the flux integral of a vector field \( \mathbf{F} \) along a curve \( C \), represented mathematically as \( \int_C \mathbf{F} \cdot \mathbf{n} \, ds \).
In this expression, \( \mathbf{n} \) is the outward-pointing unit normal vector to the curve. To compute this integral, the tangent vector to the curve \( \mathbf{r'} \) and the unit vector \( \mathbf{k} \) perpendicular to the plane are used. This ensures we are focusing only on the portion of the vector field piercing directly through the curve.
When dealing with the stream function, the flux through \( C \) becomes related to \( \psi \), and can be re-expressed in terms of the rate of change of \( \psi \) along \( C \). Thus:\[\mathbf{F} \cdot \mathbf{n} = -\frac{d\psi}{ds}\]This demonstrates that the flux integral evaluates how \( \psi \) changes as we move along the path \( C \), connecting the integral to the fundamental behavior of the streamlines defined by \( \psi \).

Path Independence

Path independence is a vital concept in vector calculus, particularly when evaluating integrals over different curves. It indicates whether the value of an integral between two points remains unchanged despite the chosen path. For flux integrals in a vector field with a stream function, path independence signifies that the flux only depends on the starting and ending points.

• In any chosen path from \( A \) to \( B \), the flux integral \( \int_C \mathbf{F} \cdot \mathbf{n} \, ds \) reduces to \( \psi(B) - \psi(A) \).
• The value of this difference is constant, meaning the integral's result does not rely on the shape of the path connecting \( A \) and \( B \).

This property simplifies calculations and shows the fundamental link between zero divergence and the lack of isolated sources or sinks inside the curve, as it confirms consistency in flow characteristics regardless of the path taken.

Zero Divergence

Zero divergence is a mathematical condition stating that a vector field does not have any net source or sink. In other words, the field is 'source-free'. It ensures the steady-state consistency of a fluid flow across space. In our problem, with the vector field \( \mathbf{F} \) being source free, it implies:

• The divergence of \( \mathbf{F} \) is zero: \( abla \cdot \mathbf{F} = 0 \).
• This results in the vector field being perfectly described by the stream function \( \psi \).

Zero divergence confirms that any changes in the vector field along a path result solely from its governing stream function. No external influences alter the field behavior, reinforcing the notion that path independence holds true for flux integrals. It underlines the connection between source-free fields and the simplification of vector calculus problems, where the field's dynamics are fully captured by its stream function.