Question

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

Short answer

The divergence is 3 and the curl is \( \langle -1, -1, -1 \rangle \).

Step-by-step solution

Understand the Concepts

The divergence of a vector field \( \vec{F} = \langle P, Q, R \rangle \) is defined as \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). The curl of a vector field is defined as \( abla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \).

Find the Divergence

For the vector field \( \vec{F} = \langle x+y, y+z, x+z \rangle \), identify \( P = x+y \), \( Q = y+z \), and \( R = x+z \). Calculate each partial derivative: \( \frac{\partial P}{\partial x} = 1 \), \( \frac{\partial Q}{\partial y} = 1 \), and \( \frac{\partial R}{\partial z} = 1 \). Then, the divergence is \( abla \cdot \vec{F} = 1 + 1 + 1 = 3 \).

Find the Curl

Calculate the components of the curl using the partial derivatives: \( \frac{\partial R}{\partial y} = 0 \), \( \frac{\partial Q}{\partial z} = 1 \), \( \frac{\partial P}{\partial z} = 0 \), \( \frac{\partial R}{\partial x} = 1 \), \( \frac{\partial Q}{\partial x} = 0 \), and \( \frac{\partial P}{\partial y} = 1 \). The curl of the vector field is \( abla \times \vec{F} = \langle 0-1, 0-1, 0-1 \rangle = \langle -1, -1, -1 \rangle \).

Key concepts

Divergence

In vector calculus, divergence measures the rate at which "stuff" expands or contracts as it moves from a point. It's like checking how air might flow out of a balloon when pricked.
It's a scalar value that provides insight into the nature of a vector field.

• Divergence of a vector field \( \vec{F} = \langle P, Q, R \rangle \) is calculated as: \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
• This simplification means checking how much the vector field is expanding or contracting in each coordinate direction.
• For our example \( \vec{F} = \langle x+y, y+z, x+z \rangle \), the divergence is \( 1 + 1 + 1 = 3 \).

This result implies a constant expansion in all directions, suggesting uniform behavior across the field.

Curl

The curl of a vector field measures the tendency to rotate around a point, much like the swirl of water down a drain.
In mathematical terms, it is a vector that describes the rotation tendency at each point.

• For a vector field \( \vec{F} = \langle P, Q, R \rangle \), the curl is given by: \( abla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \).
• Think of it as capturing the axis and intensity of rotations within the vector field.
• In our case, \( \vec{F} = \langle x+y, y+z, x+z \rangle \), the curl results in \( \langle -1, -1, -1 \rangle \).

This indicates uniform rotational symmetry around different axes, balancing out the rotational tendencies.

Vector Field

At its core, a vector field assigns a vector to every point in space, similar to how a weather map assigns a wind direction and speed at every location.
It is a function that describes physical phenomena like electromagnetic fields or fluid flows.

• A vector field \( \vec{F} = \langle P, Q, R \rangle \) has components that are functions of position \((x, y, z)\).
• Applications include physics, engineering, and environmental sciences for analyzing forces and flows.
• For our specified vector field \( \vec{F} = \langle x+y, y+z, x+z \rangle \), each vector tells us how things move or change at any location \((x, y, z)\).

Understanding vector fields enables both visualizing and predicting the behavior of complex physical systems in motion.