Question

Evaluate the three-dimensional divergence \(\nabla\). \(\mathbf{v}\) for each of the following vectors: \((\mathbf{a}) \mathbf{v}=k \hat{\mathbf{x}}\) (b) \(\mathbf{v}=k x \hat{\mathbf{x}},(\mathbf{c}) \mathbf{v}=k y \hat{\mathbf{x}} .\) We know that \(\mathbf{\nabla} \cdot \mathbf{v}\) represents the net outward flow associated with \(\mathbf{v} .\) In those cases where you found \(\nabla \cdot \mathbf{v}=0,\) make a simple sketch to illustrate that the outward flow is zero; in those cases where you found \(\nabla \cdot \mathbf{v} \neq 0,\) make a sketch to show why and whether the outflow is positive or negative.

Short answer

(a) Zero divergence; (b) Positive divergence; (c) Zero divergence.

Step-by-step solution

Understand the Divergence

The divergence of a vector field \( \mathbf{v} = (v_x, v_y, v_z) \) is given by \( abla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} \). It measures how much the vector field spreads out from a point.

Evaluate Divergence for \( \mathbf{v} = k \hat{\mathbf{x}} \)

For this vector, \( v_x = k \), and \( v_y = v_z = 0 \). Thus, the divergence is \( abla \cdot \mathbf{v} = \frac{\partial k}{\partial x} + \frac{\partial 0}{\partial y} + \frac{\partial 0}{\partial z} = 0 \) since \( k \) is a constant and its derivative is zero.

Divergence Sketch for \( \mathbf{v} = k \hat{\mathbf{x}} \)

Since \( abla \cdot \mathbf{v} = 0 \), there is no net outward flow. Picture a uniform field along the x-axis without sources or sinks, which visually shows no divergence.

Evaluate Divergence for \( \mathbf{v} = kx \hat{\mathbf{x}} \)

Here, \( v_x = kx \), so \( abla \cdot \mathbf{v} = \frac{\partial (kx)}{\partial x} = k \). Since the derivatives with respect to \( y \) and \( z \) of zero are also zero, the divergence is \( k \).

Divergence Sketch for \( \mathbf{v} = kx \hat{\mathbf{x}} \)

\( abla \cdot \mathbf{v} = k \) indicates a positive net outward flow. Visualize lines emanating from the origin along the x-axis, becoming denser as you move further out, illustrating a positive divergence.

Evaluate Divergence for \( \mathbf{v} = ky \hat{\mathbf{x}} \)

For this vector, \( v_x = ky \), yielding \( abla \cdot \mathbf{v} = \frac{\partial (ky)}{\partial x} = 0 \), because \( ky \) does not depend on \( x \). The other partial derivatives are zero as well, leading to divergence of zero.

Divergence Sketch for \( \mathbf{v} = ky \hat{\mathbf{x}} \)

With \( abla \cdot \mathbf{v} = 0 \), the outward flow is zero. Picture consistent flows along y-axis with no divergence, similar to parallel lines along the x-direction.

Key concepts

Vector Calculus

Vector Calculus is a branch of mathematics focused on vector fields, which are used to describe various physical phenomena where quantities vary over space. It involves operations like gradient, curl, and divergence, each serving to describe different properties of the vector field. Divergence is specifically concerned with how vector fields spread out from or converge into a point.

• Gradient: Measures the rate of change of a scalar field. It points in the direction of the greatest rate of increase of the field.
• Curl: Indicates the rotation or swirling strength of a vector field around a point.
• Divergence: Tells us how much a vector field spreads out from a point or converges into it. Mathematically expressed as \( abla \cdot \mathbf{v} \).

Understanding these operations helps in analyzing vector fields encountered in engineering, physics, and computer graphics.

Divergence Theorem

The Divergence Theorem, also known as Gauss's theorem, is a fundamental concept in vector calculus that connects the flow of a vector field across a closed surface to the behavior of the field within the volume enclosed by that surface. The theorem states:
\[ \int_V (abla \cdot \mathbf{v}) \ dV = \int_{S} \mathbf{v} \cdot d\mathbf{A} \]
where \( V \) is the volume, \( S \) is the closed surface, \( \mathbf{v} \) is the vector field, and \( d\mathbf{A} \) is a vector normal to the surface.

• Useful for converting volume integrals into surface integrals.
• Helps in calculating physical quantities like electric flux or fluid flow through a surface.
• Requires the vector field to be continuously differentiable over the volume.

Grasping the Divergence Theorem allows mathematicians and engineers to solve complex problems involving fluid dynamics and electromagnetism efficiently.

Vector Fields Analysis

Vector Fields Analysis entails examining vector fields to understand complex systems in various fields such as meteorology, electromagnetism, and fluid dynamics. A vector field assigns a vector to every point in space and is typically visualized using arrows. Key aspects include:

• Analyzing field lines to understand the flow direction and strength.
• Evaluating divergence to assess how much the field diverges or converges at a point.
• Observing field characteristics such as uniformity, rotation, and symmetry.

Let's consider the example of evaluating the divergence for different vector fields from the original exercise:
For \( \mathbf{v} = k \hat{\mathbf{x}} \), the divergence \( abla \cdot \mathbf{v} = 0 \), indicating no net flow out of any point, resembling parallel lines along the x-axis.
In contrast, \( \mathbf{v} = kx \hat{\mathbf{x}} \) yields \( abla \cdot \mathbf{v} = k \), illustrating a positive diverging field as it moves outward along the x-axis.
Analyzing vector fields allows one to predict and understand physical phenomena like how air circulates in weather systems or how electromagnetic fields behave around wires and circuits.