Problem 228 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\). $$ \mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j} $$

Step-by-Step Solution

Answer

The divergence of \(\mathbf{F}(x, y) = 0\).

Step 1: Understanding Divergence

Divergence is a scalar measure of how much a vector field spreads out or converges at a given point. For a 2D vector field \(\mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j}\), the divergence is given by the formula \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). Here, \(P(x, y) = x\) and \(Q(x, y) = -y\).

Step 2: Calculate Partial Derivative of P with respect to x

For \(P(x, y) = x\), the partial derivative with respect to \(x\) is: \(\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(x) = 1\). This means the rate of change of \(P\) with respect to \(x\) is 1.

Step 3: Calculate Partial Derivative of Q with respect to y

For \(Q(x, y) = -y\), the partial derivative with respect to \(y\) is: \(\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(-y) = -1\). This means the rate of change of \(Q\) with respect to \(y\) is -1.

Step 4: Calculate Divergence

Substitute the partial derivatives back into the divergence formula: \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 1 + (-1) = 0\).

Step 5: Interpret the Result

The divergence \(0\) indicates that the vector field \(\mathbf{F}(x,y) = x \mathbf{i} - y \mathbf{j}\) is neither diverging nor converging at any point; it is source-free.

Key Concepts

Vector Field

A vector field is a function that assigns a vector to every point in a space. In two dimensions, this means having a vector for each point in the plane. Each vector indicates direction and magnitude. In mathematical terms, a 2D vector field \( \mathbf{F}(x, y) \)might be represented as \( P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \), where:

• \( P(x, y) \) and \( Q(x, y) \) are scalar functions that define the components of each vector
• \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors pointing in the x and y-directions respectively

In the exercise, \( \mathbf{F}(x, y) = x \mathbf{i} - y \mathbf{j} \)specifies that at any point \( (x, y) \),the vector points \( x \)units in the x-direction and\( -y \)units in the negative y-direction. Understanding vector fields helps in visualizing how quantities like forces or velocities change over a plane.

Partial Derivatives

Partial derivatives represent the rate of change of a function with respect to one variable while keeping other variables constant. In the context of vector fields, they allow us to determine how each component of the vector changes independently.In the given example, the vector field is\( \mathbf{F}(x, y) = x \mathbf{i} - y \mathbf{j} \).To find the divergence, we took the partial derivatives:

• The partial derivative of \( P(x, y) = x \) with respect to \( x \) is \( \frac{\partial P}{\partial x} = 1 \), indicating that \( P \) changes at a uniform rate of 1 as \( x \) changes.
• The partial derivative of \( Q(x, y) = -y \) with respect to \( y \) is \( \frac{\partial Q}{\partial y} = -1 \), indicating that \( Q \) changes at a uniform rate of -1 as \( y \) changes.

Partial derivatives yield crucial information about the local behavior of the function, helping to assess changes in multi-variable functions.

Scalar Measure

Divergence in a vector field is an example of a scalar measure. It quantifies how a vector field converges or diverges at a point. Depending on its value, divergence indicates the presence or absence of sources or sinks in the field.The formula for divergence in a 2D field is \[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \]where you sum up the partial derivatives of each vector component.In our example:

• The contribution to divergence from \( P(x, y) = x \) is \( \frac{\partial P}{\partial x} = 1 \).
• The contribution from \( Q(x, y) = -y \) is \( \frac{\partial Q}{\partial y} = -1 \).
• Thus, \( abla \cdot \mathbf{F} = 1 + (-1) = 0 \), leading to a divergence of 0.

A zero divergence at all points suggests that the field is source-free; the vector field does not expand or shrink at any point. Stellar flows and electric fields often have patterns where divergence plays a key role in their analysis and understanding.