Question

Recall that if the vector field \(\mathbf{F}=\langle f, g\rangle\) is source free (zero divergence), then a stream function \(\psi\) exists such that \(f=\psi_{y}\) and \(g=-\psi_{x}\). a. Verify that the given vector field has zero divergence. b. Integrate the relations \(f=\psi_{y}\) and \(g=-\psi_{x}\) to find a stream function for the field. $$\mathbf{F}=\langle 4,2\rangle$$

Short answer

The given vector field $\mathbf{F}$ has zero divergence as the divergence is $\nabla \cdot \mathbf{F} = 0 + 0 = 0$. The corresponding stream function for this vector field is $\psi(x, y) = 4y - 2x$.

Step-by-step solution

Calculate the divergence of the vector field

To calculate the divergence of the vector field \(\mathbf{F}=\langle 4, 2\rangle\), we have to find the partial derivatives of \(f\) with respect to \(x\) and \(g\) with respect to \(y\), and then add them. The divergence of a 2D vector field is given by: $$\nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y}$$ In our case, \(f = 4\) and \(g = 2\). Let's calculate the partial derivatives: $$\frac{\partial f}{\partial x} = \frac{\partial (4)}{\partial x} = 0$$ $$\frac{\partial g}{\partial y} = \frac{\partial (2)}{\partial y} = 0$$ Now, we add them to find the divergence: $$\nabla \cdot \mathbf{F} = 0 + 0 = 0$$ The given vector field has zero divergence.

Integrate the relations to find the stream function

Since the given vector field has zero divergence, we can find a stream function \(\psi\) such that \(f=\psi_{y}\) and \(g=-\psi_{x}\). We are given \(f=4\) and \(g=2\). Let's integrate the relations to find the stream function: 1. Integrate \(f=\psi_{y}\) with respect to \(y\): $$\psi_{y} = 4$$ $$\psi(x,y) = 4y + h(x)$$ where \(h(x)\) is the integration constant, which can be a function of \(x\). 2. Integrate \(g=-\psi_{x}\) with respect to \(x\): $$-\psi_{x} = 2$$ $$\psi_{x} = -2$$ $$\psi(x,y) = -2x + k(y)$$ where \(k(y)\) is the integration constant, which can be a function of \(y\). Now, we need to combine the two expressions of \(\psi(x,y)\), which we got from integrating the two given relations. Let's equate them and find the stream function: $$4y + h(x) = -2x + k(y)$$ In order to make both the expressions equal, we choose the functions \(h(x)\) and \(k(y)\) as following: $$h(x) = -2x$$ $$k(y) = 4y$$ This gives us the final stream function: $$\psi(x,y) = 4y - 2x$$ So, the stream function for the given vector field \(\mathbf{F}=\langle 4,2\rangle\) is \(\psi(x,y) = 4y - 2x\).

Key concepts

Divergence

In vector calculus, divergence is a measure of how much a vector field spreads out from a point. For a vector field \( \mathbf{F} = \langle f, g \rangle \), divergence is calculated as the sum of the partial derivatives of its components with respect to their respective variables. It is mathematically represented as:\[abla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y}\]In simpler terms, think of divergence as a way to detect sources or sinks within the vector field. A positive divergence means there's a source, while a negative one signifies a sink. When divergence is zero, the field is source-free, indicating a balanced flow with no net production or removal of fluid in the immediate area. This concept is crucial in fluid dynamics and electromagnetism where the conservation of matter and energy needs to be analyzed.
The exercise demonstrates calculating divergence for \( \mathbf{F} = \langle 4, 2 \rangle \). Since both partial derivatives equal zero, the field has zero divergence.

Stream Function

A stream function \( \psi \) provides a powerful tool for analyzing two-dimensional, incompressible, and source-free flow in vector fields. The significance of a stream function lies in its ability to represent the flow of a fluid without directly plotting the vector field. The relations are:

• \( f = \psi_{y} \) - the partial derivative of the stream function with respect to \( y \).
• \( g = -\psi_{x} \) - the negative partial derivative of the stream function with respect to \( x \).

When a vector field is conservative and divergence-free, a stream function can be determined and used to visualize flow lines, which are always perpendicular to levels of the function. In the problem given, the stream function was found to be \( \psi(x,y) = 4y - 2x \), integrating the relations derived from the components of the vector field.
Using stream functions eliminates the need to calculate every individual flow line, simplifying the process of fluid flow analysis.

Vector Field

A vector field assigns a vector to every point in a subset of space, visually representing dynamics like fluid flow, electromagnetic fields, or gradients in physics. In this exercise, \( \mathbf{F} = \langle 4, 2 \rangle \) is a vector field in two-dimensional space. Each vector points in the same direction, showing uniform flow.
Understanding a vector field involves examining its properties through operations like divergence (indicating sources or sinks) and curl (indicating rotational features). The field \( \langle 4, 2 \rangle \) is simple and gives constant values. This means it has no variability throughout the plane and represents constant flow.
Studying vector fields allows us to interpret and predict the behavior of physical phenomena. This can help in optimizing engineering designs, improving weather predictions, and expanding scientific knowledge about natural processes.