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Chapter 17: Problem 41

Does it make sense? Are the following expressions defined? If so, state whether the result is a scalar or a vector. Assume \(\mathbf{F}\) is a sufficiently differentiable vector field and \(\varphi\) is a sufficiently differentiable scalar-valued function. a. \(\nabla \cdot \varphi\) b. \(\nabla \mathbf{F}\) c. \(\nabla \cdot \nabla \varphi\) d. \(\nabla(\nabla \cdot \varphi)\) e. \(\nabla(\nabla \times \varphi)\) f. \(\nabla \cdot(\nabla \cdot \mathbf{F})\) g. \(\nabla \times \nabla \varphi\) h. \(\nabla \times(\nabla \cdot \mathbf{F})\) i. \(\nabla \times(\nabla \times \mathbf{F})\)

Short Answer

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Question: Determine if the expressions are defined in terms of the given vector field, F, and scalar-valued function, 蠁. State whether the results are scalar or vector. a. 鈭団媴蠁 b. 鈭嘑 c. 鈭団媴鈭囅� d. 鈭�(鈭団媴蠁) e. 鈭�(鈭嚸椣�) f. 鈭団媴(鈭団媴F) g. 鈭嚸椻垏蠁 h. 鈭嚸�(鈭団媴F) i. 鈭嚸�(鈭嚸桭) Answer: a. Not defined b. Not defined c. Scalar d. Not defined e. Not defined f. Not defined g. Vector h. Not defined i. Vector

Step by step solution

01

a. \(\nabla \cdot \varphi\)#

Since the divergence operator (\(\nabla \cdot\)) acts on a vector, it is undefined when applied to a scalar function. Thus, the expression \(\nabla \cdot \varphi\) is not defined.
02

b. \(\nabla \mathbf{F}\)#

The gradient operator (\(\nabla\)) acts on scalar functions and returns a vector. However, in this case, the operand is a vector field (\(\mathbf{F}\)) rather than a scalar function. Thus, the expression \(\nabla \mathbf{F}\) is not defined.
03

c. \(\nabla \cdot \nabla \varphi\)#

First, applying the gradient operator (\(\nabla\)) to the scalar function \(\varphi\) results in a vector. Then, applying the divergence operator (\(\nabla \cdot\)) to this vector results in a scalar. Therefore, the expression \(\nabla \cdot \nabla \varphi\) is defined and results in a scalar.
04

d. \(\nabla(\nabla \cdot \varphi)\)#

As we determined in part a, the expression \(\nabla \cdot \varphi\) is not defined. Therefore, the expression \(\nabla(\nabla \cdot \varphi)\) is also not defined.
05

e. \(\nabla(\nabla \times \varphi)\)#

The curl operator (\(\nabla \times\)) acts on a vector, so it is undefined when applied to a scalar function like \(\varphi\). Thus, the expression \(\nabla \times \varphi\) is not defined, and consequently, neither is \(\nabla(\nabla \times \varphi)\).
06

f. \(\nabla \cdot(\nabla \cdot \mathbf{F})\)#

The expression \(\nabla \cdot \mathbf{F}\) is defined and results in a scalar function, but applying the divergence operator (\(\nabla \cdot\)) again to a scalar function is undefined. Hence, the expression \(\nabla \cdot(\nabla \cdot \mathbf{F})\) is not defined.
07

g. \(\nabla \times \nabla \varphi\)#

Applying the gradient operator (\(\nabla\)) to the scalar function \(\varphi\) results in a vector. Then, applying the curl operator (\(\nabla \times\)) to this vector results in a vector. Therefore, the expression \(\nabla \times \nabla \varphi\) is defined and results in a vector.
08

h. \(\nabla \times(\nabla \cdot \mathbf{F})\)#

The expression \(\nabla \cdot \mathbf{F}\) is defined and results in a scalar function. However, applying the curl operator (\(\nabla \times\)) to a scalar function is undefined. Thus, the expression \(\nabla \times(\nabla \cdot \mathbf{F})\) is not defined.
09

i. \(\nabla \times(\nabla \times \mathbf{F})\)#

Applying the curl operator (\(\nabla \times\)) to the vector field \(\mathbf{F}\) results in a vector. Then, applying the curl operator (\(\nabla \times\)) again to this vector also results in a vector. Hence, the expression \(\nabla \times(\nabla \times \mathbf{F})\) is defined and results in a vector.

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Key Concepts

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

Gradient
The **gradient** is a vector operation that measures the rate and direction of change in a scalar field. It essentially transforms a scalar function into a vector field. For a scalar function \varphi(x, y, z)\, the gradient is denoted by abla \varphi\. It is computed as the vector of partial derivatives:\ \\[ abla \varphi = \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right) \].

Here鈥檚 what you need to remember about gradients:
  • They point in the direction of the greatest rate of increase of the function.
  • The magnitude of the gradient indicates how steep the slope is at that point.
  • If you're moving along the gradient line, you're climbing or descending the slope at the steepest rate possible.
Visualize standing on a hill. The gradient tells you which direction to walk to climb the hill fastest and how steep that climb will be.
Divergence
**Divergence** is an operation that acts on a vector field to produce a scalar field. It represents the rate at which "stuff" expands from or contracts into a point. Mathematically, for a vector field \mathbf{F} = (F_x, F_y, F_z)\, the divergence is given by:

\[ abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

Think of divergence as measuring the "density" change at a point:
  • If abla \cdot \mathbf{F} > 0\, there is a source at that point (outflow).
  • If abla \cdot \mathbf{F} < 0\, there is a sink at that point (inflow).
  • If abla \cdot \mathbf{F} = 0\, the flow is constant through that point (no net change).
This is like inflating a balloon. The air pushing outward (positive divergence) inflates it, while a deflating balloon reflects negative divergence as the air rushes out.
Curl
The **curl** of a vector field measures the rotation around a point. It's specifically defined for a vector field and results in another vector field. For a vector field \mathbf{F} = (F_x, F_y, F_z)\, the curl is calculated as follows:

\[ abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \]

What does curl mean in real life? Below are some intuitive ideas:
  • A non-zero curl means there is a swirling motion.
  • The direction of the curl vector indicates the axis of rotation.
  • The magnitude tells you how strong the rotation is.
Picture water swirling down a drain. The curl helps you identify how fast and in what direction the water rotates.
Vector Field
A **vector field** is a function that assigns a vector to every point in a space. Imagine having arrows all over a surface, where each arrow represents the magnitude and direction at that point. Vector fields are commonly used to model force fields, velocity fields of fluids, and electromagnetic fields.

Characteristics of vector fields include:
  • Each point in the space has a unique vector associated with it.
  • They can be two-dimensional, like a flow on a plane, or three-dimensional, like wind through space.
  • Mathematical operations like gradient, divergence, and curl can be applied to vector fields to gain more insight into their properties.
Simply visualize a weather map with arrows showing wind speed and direction; this is an example of a vector field, illustrating both the direction and strength of the wind across the map.

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Most popular questions from this chapter

Flux across a sphere Consider the radial field \(\mathbf{F}=\langle x, y, z\rangle\) and let \(S\) be the sphere of radius \(a\) centered at the origin. Compute the outward flux of \(\mathbf{F}\) across \(S\) using the representation \(z=\pm \sqrt{a^{2}-x^{2}-y^{2}}\) for the sphere (either symmetry or two surfaces must be used).

Sketch a two-dimensional vector ficld that has zero curl everywhere in the plane.

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and a parametric description of the sphere.

Evaluating line integrals Use the given potential function \(\varphi\) of the gradient field \(\mathbf{F}\) and the curve C to evaluate the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) in two ways. a. Use a parametric description of C and evaluate the integral directly. b. Use the Fundamental Theorem for line integrals. $$\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right) / 2 ; C: \mathbf{r}(t)=\langle\cos t, \sin t, t / \pi\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Green's Theorem, flux form Consider the following regions \(R\) and vector fields \(\mathbf{F}\). a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. \(\mathbf{F}=(x,-3 y) ; R\) is the triangle with vertices (0,0),(1,2) and (0,2)
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