Question

Find the divergence and curl of the given vector field. $$ \vec{F}=\left\langle-y^{2}, x\right\rangle $$

Short answer

The divergence is 0, and the curl is \(1 + 2y\).

Step-by-step solution

Understand the Vector Field

The vector field given is \( \vec{F} = \langle -y^2, x \rangle \). This means at any point \((x, y)\), the x-component of the vector field is \(-y^2\) and the y-component is \(x\).

Find the Divergence

The divergence of a vector field \( \vec{F} = \langle P, Q \rangle \) in two dimensions is given by \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). Here, \(P = -y^2\) and \(Q = x\). Calculate:\[ \frac{\partial P}{\partial x} = \frac{\partial (-y^2)}{\partial x} = 0 \]\[ \frac{\partial Q}{\partial y} = \frac{\partial x}{\partial y} = 0 \]Thus, the divergence is: \[ abla \cdot \vec{F} = 0 + 0 = 0 \]

Find the Curl

In two dimensions for \( \vec{F} = \langle P, Q \rangle \), the curl is found as \( abla \times \vec{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \). Calculate:\[ \frac{\partial Q}{\partial x} = \frac{\partial x}{\partial x} = 1 \]\[ \frac{\partial P}{\partial y} = \frac{\partial (-y^2)}{\partial y} = -2y \]Thus, the curl is:\[ abla \times \vec{F} = 1 - (-2y) = 1 + 2y = 1 + 2y \]

Key concepts

Divergence

In vector calculus, divergence is an essential tool that helps us understand how a vector field behaves at each point in space. If you imagine a vector field as representing fluid flow, the divergence tells you if more fluid is entering or leaving a point. It measures the density of the outward flux of a vector field at a given point. For a two-dimensional vector field \( \vec{F} = \langle P, Q \rangle \), the divergence is calculated by taking the partial derivative of the first component \( P \) with respect to \( x \) and the partial derivative of the second component \( Q \) with respect to \( y \). Then, you sum these two partial derivatives:

• Divergence formula: \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \).
• It tells us about the sources or sinks within the field.
• If divergence is positive at a point, the point acts like a source.
• If it is negative, the point acts as a sink.

In the given exercise, since both partial derivatives are zero, the vector field's divergence is zero, suggesting the field is neither a source nor a sink at any point.

Curl

The concept of curl is equally fundamental in vector calculus, especially when we want to examine the rotation within a vector field. Curl provides a way to measure how much and in what manner a vector field "curls" or spins around a point. For a two-dimensional vector field \( \vec{F} = \langle P, Q \rangle \), the curl is computed as follows:

• Curl formula: \( abla \times \vec{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).
• The curl helps us identify rotational patterns.
• A zero curl indicates no rotation is present in the field.
• A non-zero curl signifies that the field curves around points where it is positive or negative.

In this exercise, the curl of the vector field \( \vec{F} = \langle -y^2, x \rangle \) results in a formula \( abla \times \vec{F} = 1 + 2y \), which means the field has varying amounts of curl based on the value of \( y \).

Vector Field

A vector field is a function that assigns a vector to every point in a space, showing how a vector quantity varies in the field. These fields are pervasive in physical and mathematical contexts, portraying things like fluid flow or electromagnetic fields. The given vector field \( \vec{F} = \langle -y^2, x \rangle \) assigns each point \( (x, y) \) to the vector \( \langle -y^2, x \rangle \):

• Each component of the field function tells us about the influence in specific directions (x and y in this case).
• Vector fields are vital in visualizing how a system behaves across different points.
• In physics, they describe how forces act on points in space or how fields like gravitational, magnetic, or electric fields exert influence.
• They can represent real-world phenomena like the direction of winds or the flow of water currents.

Understanding vector fields helps us not only to calculate divergence and curl but also to predict patterns and interactions within the field. In the exercise, the distribution of vectors across the plane provides a framework to derive its divergence and curl.