Question

Berechnen Sie von den Vektorfeldern \(\vec{F}\) die Divergenz. Geben Sie an, wo Quellen und Senken liegen, bzw. wo das Feld quellen- und senkenfrei ist. a) \(\vec{F}(x, y, z)=(x-a, y, z)\) b) \(\vec{F}(x, y, z)=\left(a,-x, z^{2}\right)\)

Short answer

a) The vector field \(\vec{F}(x, y, z)=(x-a, y, z)\) is a source everywhere. b) The vector field \(\vec{F}(x, y, z)=\left(a,-x, z^{2}\right)\) is a source when \(z > 0\), a sink when \(z < 0\), and is divergence-free when \(z = 0\).

Step-by-step solution

Compute Divergence of Vector Field a

To compute the divergence of the vector field \(\vec{F}(x, y, z)=(x-a, y, z)\), apply the formula for divergence in Cartesian coordinates: \(\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\). Substituting the given vector field components results in: \(\nabla \cdot \vec{F} = \frac{\partial (x-a)}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 0 + 0 = 1\).

Determine Nature of Vector Field a

Since the divergence of vector field a is positive and constant for all points, this vector field serves as a source everywhere.

Compute Divergence of Vector Field b

To compute the divergence of the vector field \(\vec{F}(x, y, z)=(a, -x, z^2)\), use the formula for divergence in Cartesian coordinates as above. Substituting the given vector field components results in: \(\nabla \cdot \vec{F} = \frac{\partial a}{\partial x} - \frac{\partial x}{\partial y} + \frac{\partial z^2}{\partial z} = 0 - 0 + 2z = 2z\).

Determine Nature of Vector Field b

Since the divergence of vector field b, \(2z\), changes its sign depending on the \(z\)-value, this vector field serves as a source when \(z > 0\), a sink when \(z < 0\) and it's divergence-free when \(z = 0\).

Key concepts

Vector Calculus

Vector calculus is an essential branch of mathematics that deals with vector fields, which are functions assigning vectors to each point in a space. It's like giving every speck of space its own direction and magnitude using vectors.

A significant part of vector calculus involves understanding concepts like gradient, divergence, and curl. These are tools to analyze vector fields and understand their behaviors. Gradient describes the rate and direction of change in a scalar field, while divergence and curl deal with vector fields.

Divergence, in particular, is a measure of a vector field's tendency to converge into or away from a point. If you imagine how air flows through the room as cool air from your air conditioner spreads out, divergence would tell you if air seems to magically appear or disappear at any point. Calculating divergence involves using partial derivatives, a tool that helps find rates of change in multi-dimensional spaces.

In the field of vector calculus, finding divergence helps in identifying "sources" and "sinks" of a field. The mathematical notation for divergence uses the del operator \(abla\), showing how the field spreads out in all three dimensions; for example, in Cartesian coordinates, it is expressed as \( abla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \). Understanding these concepts allows for solving more complex problems like finding electromagnetic fields, fluid flow, or any field with substance and movement.

Sources and Sinks

Sources and sinks in vector fields relate to the divergence we've previously discussed. When thinking of these, consider what would happen if you added or removed fluid at different points within a flow.

A "source" is a region where a vector field spreads out, effectively producing more of whatever the field represents. Imagine a water fountain with water spraying in all directions; this represents a source. In mathematical terms, when the divergence \( abla \cdot \vec{F} \) is positive, the vector field behaves like a source.

On the other hand, a "sink" is an area where the field seems to disappear or converge into a point, like water spiraling down a drain. If the divergence is negative, the field acts like a sink.

Sometimes, a field is neither a source nor a sink—when the divergence equals zero, the field is considered "source-free" or "sink-free." This scenario is like a calm, steady flow where the amount of substance entering a region equals the amount leaving it, resulting in no creation or disappearance of the substance.

Partial Derivatives

Partial derivatives are vital tools in calculus for understanding how functions change in multi-dimensional spaces. Unlike regular derivatives, which measure the rate of change with respect to a single variable, partial derivatives look at changes in respect to one variable while keeping others constant.

This concept is crucial when dealing with functions of multiple variables, like vector fields, because it allows us to examine how each component of the field behaves independently.

For instance, when calculating the divergence of a vector field, you'll need to determine the partial derivative of each component vector. For a vector field \(\vec{F}(x, y, z)\), the divergence formula \( abla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \) shows the importance of understanding how each vector part changes over its respective axis.

To compute these partial derivatives, treat each variable function separately as if other variables are constants. This method gives clarity into the nature of vector fields, helping us explore real-world scenarios like fluid dynamics or electromagnetic fields by precisely analyzing what happens at every coordinate point.