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Chapter 26: Problem 6

A vector field \(\mathbf{F}\) is given by $$ \mathbf{F}=x^{3} y \mathbf{i}+2 y^{2} \mathbf{j}+\left(x+z^{2}\right) \mathbf{k} $$ (a) Find \(\nabla \times \mathbf{F}\). (b) State \(3 \mathbf{F}\). (c) Find \(\nabla \times(3 \mathbf{F})\). (d) Is \(3(\nabla \times \mathbf{F})\) the same as \(\nabla \times(3 \mathbf{F})\) ?

Short Answer

Expert verified
Yes, \( 3(\nabla \times \mathbf{F}) = \nabla \times (3 \mathbf{F}) \).

Step by step solution

01

Calculate the Curl of \( \mathbf{F} \)

To find the curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), use the formula \( abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \). Here, \( P = x^3 y \), \( Q = 2y^2 \), \( R = x + z^2 \). Compute the partial derivatives:\( \frac{\partial R}{\partial y} = 0 \), \( \frac{\partial Q}{\partial z} = 0 \) (since \( Q \) does not depend on \( z \)),\( \frac{\partial P}{\partial z} = 0 \), \( \frac{\partial R}{\partial x} = 1 \),\( \frac{\partial Q}{\partial x} = 0 \), \( \frac{\partial P}{\partial y} = x^3 \).Thus, \( abla \times \mathbf{F} = (0 - 0) \mathbf{i} + (0 - 1) \mathbf{j} + (0 - x^3) \mathbf{k} = -\mathbf{j} - x^3 \mathbf{k} \).
02

State \( 3 \mathbf{F} \)

To find \( 3 \mathbf{F} \), simply multiply each component of \( \mathbf{F} \) by 3:\( 3 \mathbf{F} = 3(x^3 y \mathbf{i} + 2y^2 \mathbf{j} + (x + z^2) \mathbf{k}) = 3x^3 y \mathbf{i} + 6y^2 \mathbf{j} + (3x + 3z^2) \mathbf{k} \).
03

Calculate the Curl of \( 3 \mathbf{F} \)

Now, calculate \( abla \times (3 \mathbf{F}) \). Using the same components method:For \( 3 \mathbf{F} \), \( P = 3x^3 y \), \( Q = 6y^2 \), \( R = 3x + 3z^2 \).Compute the partial derivatives:\( \frac{\partial R}{\partial y} = 0 \), \( \frac{\partial Q}{\partial z} = 0 \),\( \frac{\partial P}{\partial z} = 0 \), \( \frac{\partial R}{\partial x} = 3 \),\( \frac{\partial Q}{\partial x} = 0 \), \( \frac{\partial P}{\partial y} = 3x^3 \).Hence, \( abla \times (3 \mathbf{F}) = (0 - 0) \mathbf{i} + (0 - 3) \mathbf{j} + (0 - 3x^3) \mathbf{k} = -3 \mathbf{j} - 3x^3 \mathbf{k} \).
04

Compare \( 3(\nabla \times \mathbf{F}) \) and \( \nabla \times (3 \mathbf{F}) \)

Now compute and compare both expressions:* From Step 1, we have \( abla \times \mathbf{F} = -\mathbf{j} - x^3 \mathbf{k} \). Multiply by 3: \( 3(abla \times \mathbf{F}) = 3(-\mathbf{j} - x^3 \mathbf{k}) = -3 \mathbf{j} - 3x^3 \mathbf{k} \).* From Step 3, we have \( abla \times (3 \mathbf{F}) = -3 \mathbf{j} - 3x^3 \mathbf{k} \).Both expressions match, so \( 3(abla \times \mathbf{F}) \) is the same as \( abla \times (3 \mathbf{F}) \).

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Key Concepts

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

Curl of a Vector Field
The concept of the **curl** of a vector field is fundamental in vector calculus, often used to describe the rotation of a field around a point in three-dimensional space. Imagine a tiny paddle wheel placed at a point in the field; the curl reveals how quickly and in which direction this paddle wheel would rotate thanks to the surrounding flow. For a vector field \( \mathbf{F} \), represented as \( P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl \( abla \times \mathbf{F} \) can be computed using:
  • \( \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} \)
  • \( \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} \)
  • \( \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \)
The resulting vector provides a comprehensive measure of the rotation, or 'twisting', happening at each point of the vector field. A field with zero curl is often described as "irrotational," meaning no such twisting occurs.
Partial Derivatives
**Partial derivatives** are a key tool in vector calculus, used to understand how functions change as individual variables are varied while others are held constant. They play a crucial role when calculating the curl of a vector field, as seen in each component of the curl formula.To compute the partial derivative, you consider only change with respect to one variable at a time. For example:
  • The partial derivative \( \frac{\partial P}{\partial y} \) considers how the component \( P \) of the vector field changes as \( y \) changes, with other variables fixed.
  • Similarly, \( \frac{\partial R}{\partial x} \) shows how \( R \) changes with respect to \( x \).
Finding these derivatives helps in getting to precise components of more complex vector calculus operations like curl.
Vector Field Operations
Understanding **vector field operations** is crucial as these operations like scaling or computing the curl help analyze dynamic systems in physics and engineering. Operations on vector fields, such as scaling by a constant and taking the curl, reveal different aspects of the field's behavior.For example:
  • Scaling: Taking a vector field \( \mathbf{F} \) and creating \( c \mathbf{F} \), where \( c \) is a constant, each component of the vector is simply multiplied by \( c \). This operation maintains the field's direction but alters its magnitude.
  • Curl Analysis: Following scaling, computing the curl of the resultant vector field \( \mathbf{F} \) results in \( abla \times (c \mathbf{F}) = c(abla \times \mathbf{F}) \), showing that scaling does not affect rotational characteristics but only magnifies or reduces them by \( c \).
These operations help in manipulating the vector field to better understand and visualize the flow and rotational tendencies around regions, being exceptionally useful in electromagnetics, fluid dynamics, and other vector calculus applications.

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Most popular questions from this chapter

For any vector field \(\mathbf{F}\) and any scalar constant \(k\) is \(\nabla \cdot(k \mathbf{F})\) the same as \(k \nabla \cdot \mathbf{F} ?\)

If \(\mathbf{A}=3 y z \mathbf{i}+2 x y \mathbf{j}+x y z \mathbf{k}\) find \(\nabla \cdot \mathbf{A}\)

Suppose \(\mathbf{F}=P(x, y) \mathbf{i}+Q(x, y) \mathbf{j}\) is a two-dimensional vector field. Show that \(\mathbf{F}\) is irrotational if \(\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}\).

Scalar fields \(\phi_{1}\) and \(\phi_{2}\) are given by $$ \phi_{1}=2 x y+y^{2} z \quad \phi_{2}=x^{2} z $$ (a) Find \(\nabla \phi_{1}\). (b) Find \(\nabla \phi_{2}\). (c) State \(\phi_{1} \phi_{2}\). (d) Find \(\nabla\left(\phi_{1} \phi_{2}\right)\). (e) Find \(\phi_{1} \nabla \phi_{2}+\phi_{2} \nabla \phi_{1}\). (f) What do you conclude from (d) and (e)?

If \(\phi=x^{2} y z^{3}\), find (a) \(\nabla \phi,(\mathrm{b}) \nabla \phi\) at \((1,2,1),(\mathrm{c})|\nabla \phi|\) at \((1,2,1)\).
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