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Chapter 16: Problem 12

Let \(f\) be a scalar field and \(\mathbf{F}\) a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field. \(\begin{array}{ll}{\text { (a) curl } f} & {\text { (b) grad } f} \\ {\text { (c) div } \mathbf{F}} & {\text { (d) curl(grad } f} \\ {\text { (e) grad } \mathbf{F}} & {\text { (f) grad(div } \mathbf{F}}\end{array}\) \((\mathrm{g})\) div(grad \(f ) \quad\) (h) grad(div \(f )\) (i) curl(curl \(\mathbf{F} ) \quad(\) j) div(div \(\mathbf{F})\) (k) (grad \(f ) \times(\) div \(\mathbf{F}) \quad\) (1) div(curl(grad \(f ) )\)

Short Answer

Expert verified
Expressions (b), (c), (g), and (i) are meaningful.

Step by step solution

01

Analyze curl f

The curl of a scalar field is undefined or not meaningful because curl is an operator applied to vector fields. Since \( f \) is a scalar field, \( \text{curl} \, f \) is not meaningful.
02

Analyze grad f

The gradient of a scalar field \( f \) is a vector field. Therefore, \( \text{grad} \, f \) is meaningful and results in a vector field.
03

Analyze div F

The divergence of a vector field \( \mathbf{F} \) is a scalar field. Therefore, \( \text{div} \, \mathbf{F} \) is meaningful and results in a scalar field.
04

Analyze curl(grad f)

The curl of the gradient of a scalar field is not meaningful. The operation results in a vector field that is always zero, which is often considered not useful or interesting.
05

Analyze grad F

The gradient of a vector field is not a standard operation and is not meaningful in typical contexts. Therefore, \( \text{grad} \, \mathbf{F} \) is not meaningful.
06

Analyze grad(div F)

The gradient of the divergence of a vector field \( \mathbf{F} \) is a meaningful operation and results in a vector field.
07

Analyze div(grad f)

The divergence of the gradient of a scalar field \( f \) is a meaningful operation and results in a scalar field, often referred to as the Laplacian \( abla^2 f \).
08

Analyze grad(div f)

The divergence of a scalar field is not meaningful, hence \( \text{grad(div } f) \) is not meaningful because 'div \( f \)' would not be defined.
09

Analyze curl(curl F)

The curl of the curl of a vector field \( \mathbf{F} \) is meaningful and results in another vector field.
10

Analyze div(div F)

The divergence of the divergence of a vector field is not meaningful in a standard sense because it requires two subsequent divergence operations which are not applicable.
11

Analyze (grad f) \times (div F)

The operation involves taking the cross product between a vector field \( \text{grad} \, f \) and a scalar \( \text{div} \, \mathbf{F} \), which is not defined. Consequently, this is not meaningful.
12

Analyze div(curl(grad f))

Combining these operations ultimately encounters the same static zero from \( \text{curl(grad } f) \), and since this is without divergence, it becomes not meaningful.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Field
In vector calculus, a **scalar field** assigns a single scalar value to every point in space. This could be anything like temperature or pressure at a point. Think of it like coloring a map with different colors: every point on the map might have a different temperature, represented by a different color.

Some important points to remember about scalar fields:
  • Scalar fields are quantities that vary, but they don't have a direction.
  • A single value is assigned to every point in the field.
  • Examples include temperature distribution, population density, or elevation on a map.
The scalar field can be denoted by functions like \( f(x, y, z) \), where each input point in space has a corresponding scalar value.
Vector Field
A **vector field** is quite different from a scalar field. It assigns a vector to each point in space. Imagine the wind: at every point, there is both a speed and a direction.

Some important features of vector fields are:
  • They associate a vector with every point in space, providing both magnitude and direction.
  • Common examples include magnetic fields, velocity fields in fluid flow, and force fields.
  • Mathematically, a vector field can be represented as \( \mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k} \).
Understanding vector fields is crucial for describing phenomena where direction and magnitude both matter, like in physics.
Curl
The **curl** operation applies to vector fields to measure the rotation at a point in the field. If you imagine a bug moving along a vector field, curl helps to understand if the bug experiences rotational motion.

Key points about curl include:
  • The curl, applied to a vector field, results in another vector field.
  • It indicates the rotational tendency at a point in the field.
  • If the curl is zero, the field has no rotational effect at that point.
  • Curl is often expressed as \( abla \times \mathbf{F} \), where \( \mathbf{F} \) is the vector field.
Curl is vital in fluid dynamics and electromagnetism, where it helps analyze field rotations.
Gradient
The **gradient** of a scalar field indicates the direction and rate of fastest increase of that field. Imagine climbing a hill, the gradient shows where the slope is steepest.

Characteristics of the gradient include:
  • Applying the gradient to a scalar field results in a vector field.
  • The vector field points in the direction of the greatest rate of increase of the scalar field.
  • Mathematically, it is noted as \( abla f \) for a scalar field \( f \).
The gradient is a powerful concept in illustrating changes across a spatial field, very useful in both physics and engineering contexts.
Divergence
**Divergence** measures the magnitude of a source or sink at a particular point in a vector field. It tells us whether a point is acting like a source, a sink, or neither within the vector field.

Important features of divergence are:
  • Divergence applied to a vector field yields a scalar field.
  • It captures the degree to which a point acts as a source (positive divergence) or sink (negative divergence).
  • Mathematically represented as \( abla \cdot \mathbf{F} \), where \( \mathbf{F} \) is the vector field.
  • If divergence is zero, it means the point behaves neutrally, neither being a source nor a sink.
Divergence plays a crucial role in vector calculus, especially in understanding fluid and electromagnetic fields.

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Most popular questions from this chapter

Find the work done by the force field \(\mathbf{F}(x, y)=x \mathbf{i}+(y+2) \mathbf{j}\) in moving an object along an arch of the cycloid \(\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}, 0 \leqslant t \leqslant 2 \pi\)

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\(37-47\) Find the area of the surface. The part of the plane \(3 x+2 y+z=6\) that lies in the first octant
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