Question

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

Short answer

The divergence is \( 2x + 2y + 2z \), and the curl is \( \langle 2y, 2z, 2x \rangle \).

Step-by-step solution

Identifying the Vector Field Components

To begin solving the problem, identify the components of the vector field \( \vec{F} = \langle x^2 + z^2, x^2 + y^2, y^2 + z^2 \rangle \). These components can be labeled as \( F_1 = x^2 + z^2 \), \( F_2 = x^2 + y^2 \), and \( F_3 = y^2 + z^2 \).

Calculating the Divergence

The divergence of a vector field \( \vec{F} = \langle F_1, F_2, F_3 \rangle \) is given by the formula \( abla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \). Calculate each partial derivative: \( \frac{\partial F_1}{\partial x} = 2x \), \( \frac{\partial F_2}{\partial y} = 2y \), and \( \frac{\partial F_3}{\partial z} = 2z \). Add these to get the divergence: \( abla \cdot \vec{F} = 2x + 2y + 2z \).

Setting Up for Curl Calculation

The curl of a vector field \( \vec{F} \) is given by \( abla \times \vec{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \). Identify the relevant partial derivatives needed for each component of the curl.

Calculating Each Component of the Curl

Compute each component of the curl:- For the x-component: \( \frac{\partial F_3}{\partial y} = 2y \) and \( \frac{\partial F_2}{\partial z} = 0 \), thus the x-component is \( 2y - 0 = 2y \).- For the y-component: \( \frac{\partial F_1}{\partial z} = 2z \) and \( \frac{\partial F_3}{\partial x} = 0 \), thus the y-component is \( 2z - 0 = 2z \).- For the z-component: \( \frac{\partial F_2}{\partial x} = 2x \) and \( \frac{\partial F_1}{\partial y} = 0 \), thus the z-component is \( 2x - 0 = 2x \).

Writing the Curl Vector

Combine the components calculated into the curl vector: \( abla \times \vec{F} = \langle 2y, 2z, 2x \rangle \).

Key concepts

Understanding Divergence

The concept of divergence helps us measure how much a vector field spreads out or converges at a given point. Imagine it as the net flow of the field's vectors per unit volume outward from a small region around the point. If you picture a field of arrows representing wind speed and direction, divergence at a point tells you how air is "created" or "vanished" there.

• A positive divergence indicates a source; think of water gushing out from a spring.
• A negative divergence indicates a sink, where material is being "sucked in."

To calculate divergence mathematically, for a vector field \( \vec{F} = \langle F_1, F_2, F_3 \rangle \), you use the formula \( abla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \).
In the given exercise, the divergence was computed to be \( 2x + 2y + 2z \), indicating how the vector field expands or contracts at each point \( (x, y, z) \). This result shows a uniformly increasing divergence as we move away from the origin in any spatial direction.

Exploring the Curl

Curl measures the rotational tendency or the "twisting" motion of a vector field around a point. If you've ever watched water swirling around a drain, you can relate that twisting motion to the concept of curl.

• A non-zero curl at a point means there's some rotation occurring at that point.
• If the curl is zero, the vector field does not circulate at that point, indicating a more straightforward, linear flow or uniform direction.

Calculating the curl involves taking the cross product of the del operator \( abla \) and the vector field \( \vec{F} \) to produce a new vector that represents rotational behavior. For the vector field \( \vec{F} = \langle F_1, F_2, F_3 \rangle \), the formula is \( abla \times \vec{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \).
For our specific example, the curl was determined to be \( \langle 2y, 2z, 2x \rangle \), indicating that the vector field does impart a rotational effect in three-dimensional space.

Defining a Vector Field

A vector field is a mathematical construction where each point in a space is associated with a vector. You can think of it as representing various physical quantities such as velocity, force, or electromagnetic fields.

• In 2D or 3D spaces, vector fields are usually visualized as arrows - each with a direction and magnitude.
• They can describe the behavior of a quantity specified at each point, influencing how we model natural phenomena.

When analyzing vector fields, we're often interested in understanding how they diverge or curl, as these tell us about the field's expansion or rotational properties.
In the exercise, you encountered a 3D vector field \( \vec{F} = \langle x^2 + z^2, x^2 + y^2, y^2 + z^2 \rangle \). Each part of the vector field corresponds to the components along the \( x \), \( y \), and \( z \) axes. Mastering vector fields means being able to visualize these arrows and apply calculus to unpack the field's properties, thus predicting how they behave in complex environments.